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Research Article On Simultaneous Approximation of Modified...
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Research ArticleOn Simultaneous Approximation of ModifiedBaskakov-Durrmeyer Operators
Prashantkumar G Patel12 and Vishnu Narayan Mishra13
1Department of Applied Mathematics and Humanities Sardar Vallabhbhai National Institute of TechnologyIchchhanath Mahadev Dumas Road Surat Gujarat 395 007 India2Department of Mathematics St Xavier College Ahmedabad Gujarat 380 009 India3L 1627 Awadh Puri Colony Beniganj Phase-III Opposite-Industrial Training Institute (ITI) Ayodhya Main Road FaizabadUttar Pradesh 224 001 India
Correspondence should be addressed to Prashantkumar G Patel prashant225gmailcom
Received 22 July 2015 Accepted 10 September 2015
Academic Editor Ying Hu
Copyright copy 2015 P G Patel and V N Mishra This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter 120574 gt 0 We compute the momentsof these modified operators Also we establish pointwise convergence Voronovskaja type asymptotic formula and an errorestimation in terms of second order modification of continuity of the function for the operators 119861120572120573
119899120574(119891 119909)
1 Introduction
For 119909 isin [0infin) 120574 gt 0 0 le 120572 le 120573 and 119891 isin 119862[0infin)we consider a certain integral type generalized Baskakovoperators as
119861120572120573
119899120574(119891 (119905) 119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891 (119899119905 + 120572
119899 + 120573)119889119905
+ 1199011198990120574
(119909) 119891(120572
119899 + 120573)
= int
infin
0
119882119899120574
(119909 119905) 119891 (119899119905 + 120572
119899 + 120573)119889119905
(1)
where
119901119899119896120574
(119909) =Γ (119899120574 + 119896)
Γ (119896 + 1) Γ (119899120574)sdot
(120574119909)119896
(1 + 120574119909)(119899120574)+119896
119887119899119896120574
(119905) =120574Γ (119899120574 + 119896 + 1)
Γ (119896) Γ (119899120574 + 1)sdot
(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119882119899120574
(119909 119905) =
infin
sum
119896=1
119901119899119896120574
(119909) 119887119899119896120574
(119905) + (1 + 120574119909)minus119899120574
120575 (119905)
(2)
120575(119905) being the Dirac delta functionThe operators defined by (1) are the generalization of
the integral modification of well-known Baskakov operatorshaving weight function of some beta basis function As aspecial case that is 120574 = 1 the operators (1) reduce to theoperators very recently studied in [1 2] Inverse results ofsame type of operatorswere established in [3] Also if120572 = 120573 =
0 the operators (1) reduce to the operators recently studied in[4] and if 120572 = 120573 = 0 and 120574 = 1 the operators (1) reduce to theoperators studied in [5] The 119902-analog of the operators (1) isdiscussed in [6] We refer to some of the important papers onthe recent development on similar type of the operators [7ndash9]
Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2015 Article ID 805395 10 pageshttpdxdoiorg1011552015805395
2 International Journal of Analysis
The present a paper that deals with the study of simultaneousapproximation for the operators 119861120572120573
119899120574
2 Moments and Recurrence Relations
Lemma 1 If one defines the central moments for every119898 isin Nas
120583119899119898120574
(119909) = 119861120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
(3)
then 1205831198990120574
(119909) = 1 1205831198991120574
(119909) = (120572minus120573119909)(119899+120573) and for 119899 gt 120574119898one has the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909)
sdot 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
+ 119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)
sdot 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)
sdot 120583119899119898minus1120574
(119909)
(4)
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) one has 120583
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
) where [120572]
denotes the integral part of 120572
Proof Taking derivative of the above
120583(1)
119899119898120574(119909) = minus119898
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905 minus 1198981199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898minus1
+
infin
sum
119896=1
119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119901(1)
1198990120574(119909)
sdot (120572
119899 + 120573minus 119909)
119898
= minus119898120583119899119898minus1120574
(119909) +
infin
sum
119896=1
119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119901(1)
1198990120574(119909)
sdot (120572
119899 + 120573minus 119909)
119898
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
=
infin
sum
119896=1
119909 (1 + 120574119909) 119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119909 (1 + 120574119909)
sdot 119901(1)
1198990120574(119909) (
120572
119899 + 120573minus 119909)
119898
(5)
Using 119909(1 + 120574119909)119901(1)
119899119896120574(119909) = (119896 minus 119899119909)119901
119899119896120574(119909) we get
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
=
infin
sum
119896=1
(119896 minus 119899119909) 119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + (minus119899119909) 1199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898
=
infin
sum
119896=1
119896119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 minus 119899119909120583119899119898120574
(119909) = 119868
minus 119899119909120583119899119898120574
(119909)
(6)
We can write 119868 as
119868 =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
(119896 minus 1) minus (119899 + 2120574) 119905 119887119899119896120574
(119905)
sdot (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 +
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905)
sdot (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + (119899 + 2120574)
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119905119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 = 1198681
+ 1198682
(say)
(7)
To estimate 1198682using 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus
(120572(119899 + 120573) minus 119909) we have
1198682=
(119899 + 2120574) (119899 + 120573)
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 minus (120572
119899 + 120573
minus 119909)
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
International Journal of Analysis 3
=(119899 + 2120574) (119899 + 120573)
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + 1199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898+1
minus (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
=(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909) minus (120572
119899 + 120573minus 119909)
sdot 120583119899119898120574
(119909)
(8)
Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +
2120574)119905119887119899119896120574
(119905) = 119905(1 + 120574119905)119887(1)
119899119896120574(119905) we have
1198681=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119905119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 120574
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
1199052
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= 1198691+ 1198692
(say)
(9)
Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +
120573) minus 119909) we get
1198691=
119899 + 120573
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
(10)
Now integrating by parts we get
1198691= minus (119898 + 1)
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= minus (119898 + 1)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
+ 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898minus1
+
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 1199011198990120574
(119909) (120572
119899 + 120573
minus 119909)
119898
1198691= minus119898120583
119899119898120574(119909) + 119898(
120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(11)
Proceeding in the similar manner we obtain the estimate 1198692
as
1198692= minus
120574 (119899 + 120573) (119898 + 2)
119899120583119899119898+1120574
(119909)
+ 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
120583119899119898minus1120574
(119909)
(12)
Combining (6)ndash(12) we get
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909) = minus119898120583
119899119898120574(119909)
+ 119898(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909) minus
120574 (119899 + 120573) (119898 + 2)
119899
sdot 120583119899119898+1120574
(119909) + 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)
4 International Journal of Analysis
sdot 120583119899119898120574
(119909) minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
sdot 120583119899119898minus1120574
(119909) minus 119899119909120583119899119898120574
(119909)
+(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909)
minus (120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
(13)
Hence
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909) 120583(1)
119899119898120574(119909)
+ 119898120583119899119898minus1120574
(119909) + 119898119899 + 1198992
119909
minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(14)
This completes the proof of Lemma 1
Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as
119880119899119898120574
(119909) =
infin
sum
119896=0
119901119899119896120574
(119909) (119896
119899minus 119909)
119898
(15)
then 1198801198990120574
(119909) = 1 1198801198991120574
(119909) = 0 and 119899119880119899119898+1120574
(119909) = 119909(1 +
120574119909)(119880(1)
119899119898120574(119909) + 119898119880
119899119898minus1120574(119909))
Consequently for all 119909 isin [0infin) we have 119880119899119898120574
(119909) =
119874(119899minus[(119898+1)2]
)
Remark 3 It is easily verified from Lemma 1 that for each 119909 isin
[0infin)
119861120572120573
119899120574(119905119898
119909) =119899119898
Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119909119898
+119898119899119898minus1
Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)
+ 120572(119899
120574minus 119898 + 1)119909
119898minus1
+120572119898 (119898 minus 1) 119899
119898minus2
Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898
minus 2) +120572 (119899120574 minus 119898 + 2)
2119909119898minus2
+ 119874 (119899minus2
)
(16)
Lemma 4 (see [10]) The polynomials119876119894119895119903120574
(119909) exist indepen-dent of 119899 and 119896 such that
119909 (1 + 120574119909)119903
119863119903
[119901119899119896120574
(119909)]
= sum
2119894+119895le119903
119894119895ge0
119899119894
(119896 minus 119899119909)119895
119876119894119895119903120574
(119909) 119901119899119896120574
(119909)
where 119863 equiv119889
119889119909
(17)
Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)
= 119874(119905120592
) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has
(119861120572120573
119899120574)(119903)
(119891 119909) =119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) 119891(119903)
(119899119905 + 120572
119899 + 120573)119889119905
(18)
Proof First
(119861120572120573
119899120574)(1)
(119891 119909)
=
infin
sum
119896=1
119901(1)
119899119896120574(119909) int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(19)
Now using the identities
119901(1)
119899119896120574(119909) = 119899 119901
119899+120574119896minus1120574(119909) minus 119901
119899+120574119896120574(119909)
119887(1)
119899119896120574(119909) = (119899 + 120574) 119887
119899+120574119896minus1120574(119909) minus 119887
119899+120574119896120574(119909)
(20)
for 119896 ge 1 we have
(119861120572120573
119899120574)(1)
(119891 119909) =
infin
sum
119896=1
119899 119901119899+120574119896minus1120574
(119909) minus 119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) = 119899119901
119899+1205740120574(119909)
sdot int
infin
0
119887119899+1205741120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) + 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896+1120574
(119905) minus 119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
International Journal of Analysis 5
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
sdot int
infin
0
(119899 + 120574) (1 + 120574119905)minus119899120574minus2
119891(119899119905 + 120572
119899 + 120573)119889119905
+ 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
(minus1
119899119887(1)
119899minus120574119896+1120574(119905)) 119891(
119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(21)
Integrating by parts we get
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
+1198992
119899 + 120573(1 + 120574119909)
minus119899120574minus1
sdot int
infin
0
(1 + 120574119905)minus119899120574minus1
119891(1)
(119899119905 + 120572
119899 + 120573)119889119905 +
119899
119899 + 120573
sdot
infin
sum
119896=1
119901119899+120574119896120574
(119909) int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(119861120572120573
119899120574)(1)
(119891 119909) =119899
119899 + 120573
infin
sum
119896=0
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
(22)
Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then
(119861120572120573
119899120574)(119894)
(119891 119909) =119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119894119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(23)
Thus using the identities (20) we have
(119861120572120573
119899120574)(119894+1)
(119891 119909)
=119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
(119899
120574+ 119894)
sdot 119901119899+120574(119894+1)119896minus1120574
(119909) minus 119901119899+120574(119894+1)119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905 minus (
119899
120574+ 119894) (1 + 120574119909)
minus119899120574minus119894minus1
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus1205741198941+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
minus119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574119894119896+119894+1120574
(119905) minus 119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)1+119894120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)119896+119894+1120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(24)
Integrating by parts we obtain
(119861120572120573
119899120574)(119894+1)
(119891 119909) =119899119894+1
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894+1
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574(119894minus1)119896+119894+1120574
(119905) 119891(119894+1)
(119899119905 + 120572
119899 + 120573)119889119905
(25)
This completes the proof of Lemma 5
3 Direct Theorems
This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Analysis
The present a paper that deals with the study of simultaneousapproximation for the operators 119861120572120573
119899120574
2 Moments and Recurrence Relations
Lemma 1 If one defines the central moments for every119898 isin Nas
120583119899119898120574
(119909) = 119861120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
(3)
then 1205831198990120574
(119909) = 1 1205831198991120574
(119909) = (120572minus120573119909)(119899+120573) and for 119899 gt 120574119898one has the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909)
sdot 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
+ 119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)
sdot 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)
sdot 120583119899119898minus1120574
(119909)
(4)
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) one has 120583
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
) where [120572]
denotes the integral part of 120572
Proof Taking derivative of the above
120583(1)
119899119898120574(119909) = minus119898
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905 minus 1198981199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898minus1
+
infin
sum
119896=1
119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119901(1)
1198990120574(119909)
sdot (120572
119899 + 120573minus 119909)
119898
= minus119898120583119899119898minus1120574
(119909) +
infin
sum
119896=1
119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119901(1)
1198990120574(119909)
sdot (120572
119899 + 120573minus 119909)
119898
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
=
infin
sum
119896=1
119909 (1 + 120574119909) 119901(1)
119899119896120574(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119909 (1 + 120574119909)
sdot 119901(1)
1198990120574(119909) (
120572
119899 + 120573minus 119909)
119898
(5)
Using 119909(1 + 120574119909)119901(1)
119899119896120574(119909) = (119896 minus 119899119909)119901
119899119896120574(119909) we get
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909)
=
infin
sum
119896=1
(119896 minus 119899119909) 119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + (minus119899119909) 1199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898
=
infin
sum
119896=1
119896119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 minus 119899119909120583119899119898120574
(119909) = 119868
minus 119899119909120583119899119898120574
(119909)
(6)
We can write 119868 as
119868 =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
(119896 minus 1) minus (119899 + 2120574) 119905 119887119899119896120574
(119905)
sdot (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 +
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905)
sdot (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + (119899 + 2120574)
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119905119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 = 1198681
+ 1198682
(say)
(7)
To estimate 1198682using 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus
(120572(119899 + 120573) minus 119909) we have
1198682=
(119899 + 2120574) (119899 + 120573)
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 minus (120572
119899 + 120573
minus 119909)
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
International Journal of Analysis 3
=(119899 + 2120574) (119899 + 120573)
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + 1199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898+1
minus (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
=(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909) minus (120572
119899 + 120573minus 119909)
sdot 120583119899119898120574
(119909)
(8)
Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +
2120574)119905119887119899119896120574
(119905) = 119905(1 + 120574119905)119887(1)
119899119896120574(119905) we have
1198681=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119905119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 120574
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
1199052
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= 1198691+ 1198692
(say)
(9)
Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +
120573) minus 119909) we get
1198691=
119899 + 120573
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
(10)
Now integrating by parts we get
1198691= minus (119898 + 1)
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= minus (119898 + 1)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
+ 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898minus1
+
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 1199011198990120574
(119909) (120572
119899 + 120573
minus 119909)
119898
1198691= minus119898120583
119899119898120574(119909) + 119898(
120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(11)
Proceeding in the similar manner we obtain the estimate 1198692
as
1198692= minus
120574 (119899 + 120573) (119898 + 2)
119899120583119899119898+1120574
(119909)
+ 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
120583119899119898minus1120574
(119909)
(12)
Combining (6)ndash(12) we get
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909) = minus119898120583
119899119898120574(119909)
+ 119898(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909) minus
120574 (119899 + 120573) (119898 + 2)
119899
sdot 120583119899119898+1120574
(119909) + 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)
4 International Journal of Analysis
sdot 120583119899119898120574
(119909) minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
sdot 120583119899119898minus1120574
(119909) minus 119899119909120583119899119898120574
(119909)
+(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909)
minus (120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
(13)
Hence
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909) 120583(1)
119899119898120574(119909)
+ 119898120583119899119898minus1120574
(119909) + 119898119899 + 1198992
119909
minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(14)
This completes the proof of Lemma 1
Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as
119880119899119898120574
(119909) =
infin
sum
119896=0
119901119899119896120574
(119909) (119896
119899minus 119909)
119898
(15)
then 1198801198990120574
(119909) = 1 1198801198991120574
(119909) = 0 and 119899119880119899119898+1120574
(119909) = 119909(1 +
120574119909)(119880(1)
119899119898120574(119909) + 119898119880
119899119898minus1120574(119909))
Consequently for all 119909 isin [0infin) we have 119880119899119898120574
(119909) =
119874(119899minus[(119898+1)2]
)
Remark 3 It is easily verified from Lemma 1 that for each 119909 isin
[0infin)
119861120572120573
119899120574(119905119898
119909) =119899119898
Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119909119898
+119898119899119898minus1
Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)
+ 120572(119899
120574minus 119898 + 1)119909
119898minus1
+120572119898 (119898 minus 1) 119899
119898minus2
Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898
minus 2) +120572 (119899120574 minus 119898 + 2)
2119909119898minus2
+ 119874 (119899minus2
)
(16)
Lemma 4 (see [10]) The polynomials119876119894119895119903120574
(119909) exist indepen-dent of 119899 and 119896 such that
119909 (1 + 120574119909)119903
119863119903
[119901119899119896120574
(119909)]
= sum
2119894+119895le119903
119894119895ge0
119899119894
(119896 minus 119899119909)119895
119876119894119895119903120574
(119909) 119901119899119896120574
(119909)
where 119863 equiv119889
119889119909
(17)
Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)
= 119874(119905120592
) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has
(119861120572120573
119899120574)(119903)
(119891 119909) =119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) 119891(119903)
(119899119905 + 120572
119899 + 120573)119889119905
(18)
Proof First
(119861120572120573
119899120574)(1)
(119891 119909)
=
infin
sum
119896=1
119901(1)
119899119896120574(119909) int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(19)
Now using the identities
119901(1)
119899119896120574(119909) = 119899 119901
119899+120574119896minus1120574(119909) minus 119901
119899+120574119896120574(119909)
119887(1)
119899119896120574(119909) = (119899 + 120574) 119887
119899+120574119896minus1120574(119909) minus 119887
119899+120574119896120574(119909)
(20)
for 119896 ge 1 we have
(119861120572120573
119899120574)(1)
(119891 119909) =
infin
sum
119896=1
119899 119901119899+120574119896minus1120574
(119909) minus 119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) = 119899119901
119899+1205740120574(119909)
sdot int
infin
0
119887119899+1205741120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) + 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896+1120574
(119905) minus 119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
International Journal of Analysis 5
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
sdot int
infin
0
(119899 + 120574) (1 + 120574119905)minus119899120574minus2
119891(119899119905 + 120572
119899 + 120573)119889119905
+ 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
(minus1
119899119887(1)
119899minus120574119896+1120574(119905)) 119891(
119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(21)
Integrating by parts we get
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
+1198992
119899 + 120573(1 + 120574119909)
minus119899120574minus1
sdot int
infin
0
(1 + 120574119905)minus119899120574minus1
119891(1)
(119899119905 + 120572
119899 + 120573)119889119905 +
119899
119899 + 120573
sdot
infin
sum
119896=1
119901119899+120574119896120574
(119909) int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(119861120572120573
119899120574)(1)
(119891 119909) =119899
119899 + 120573
infin
sum
119896=0
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
(22)
Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then
(119861120572120573
119899120574)(119894)
(119891 119909) =119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119894119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(23)
Thus using the identities (20) we have
(119861120572120573
119899120574)(119894+1)
(119891 119909)
=119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
(119899
120574+ 119894)
sdot 119901119899+120574(119894+1)119896minus1120574
(119909) minus 119901119899+120574(119894+1)119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905 minus (
119899
120574+ 119894) (1 + 120574119909)
minus119899120574minus119894minus1
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus1205741198941+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
minus119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574119894119896+119894+1120574
(119905) minus 119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)1+119894120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)119896+119894+1120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(24)
Integrating by parts we obtain
(119861120572120573
119899120574)(119894+1)
(119891 119909) =119899119894+1
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894+1
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574(119894minus1)119896+119894+1120574
(119905) 119891(119894+1)
(119899119905 + 120572
119899 + 120573)119889119905
(25)
This completes the proof of Lemma 5
3 Direct Theorems
This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 3
=(119899 + 2120574) (119899 + 120573)
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + 1199011198990120574
(119909)
sdot (120572
119899 + 120573minus 119909)
119898+1
minus (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
=(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909) minus (120572
119899 + 120573minus 119909)
sdot 120583119899119898120574
(119909)
(8)
Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +
2120574)119905119887119899119896120574
(119905) = 119905(1 + 120574119905)119887(1)
119899119896120574(119905) we have
1198681=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119905119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 120574
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
1199052
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= 1198691+ 1198692
(say)
(9)
Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +
120573) minus 119909) we get
1198691=
119899 + 120573
119899
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898+1
119889119905 + (120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887(1)
119899119896120574(119905) (
119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
(10)
Now integrating by parts we get
1198691= minus (119898 + 1)
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
= minus (119898 + 1)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898
+ 119898(120572
119899 + 120573minus 119909)
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898minus1
119889119905
+ 1199011198990120574
(119909) (120572
119899 + 120573minus 119909)
119898minus1
+
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
119898
119889119905 + 1199011198990120574
(119909) (120572
119899 + 120573
minus 119909)
119898
1198691= minus119898120583
119899119898120574(119909) + 119898(
120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(11)
Proceeding in the similar manner we obtain the estimate 1198692
as
1198692= minus
120574 (119899 + 120573) (119898 + 2)
119899120583119899119898+1120574
(119909)
+ 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
120583119899119898minus1120574
(119909)
(12)
Combining (6)ndash(12) we get
119909 (1 + 120574119909) 120583(1)
119899119898120574(119909) + 119898120583
119899119898minus1120574(119909) = minus119898120583
119899119898120574(119909)
+ 119898(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909) minus
120574 (119899 + 120573) (119898 + 2)
119899
sdot 120583119899119898+1120574
(119909) + 2120574(119899 + 120573) (119898 + 1)
119899(
120572
119899 + 120573minus 119909)
4 International Journal of Analysis
sdot 120583119899119898120574
(119909) minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
sdot 120583119899119898minus1120574
(119909) minus 119899119909120583119899119898120574
(119909)
+(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909)
minus (120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
(13)
Hence
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909) 120583(1)
119899119898120574(119909)
+ 119898120583119899119898minus1120574
(119909) + 119898119899 + 1198992
119909
minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(14)
This completes the proof of Lemma 1
Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as
119880119899119898120574
(119909) =
infin
sum
119896=0
119901119899119896120574
(119909) (119896
119899minus 119909)
119898
(15)
then 1198801198990120574
(119909) = 1 1198801198991120574
(119909) = 0 and 119899119880119899119898+1120574
(119909) = 119909(1 +
120574119909)(119880(1)
119899119898120574(119909) + 119898119880
119899119898minus1120574(119909))
Consequently for all 119909 isin [0infin) we have 119880119899119898120574
(119909) =
119874(119899minus[(119898+1)2]
)
Remark 3 It is easily verified from Lemma 1 that for each 119909 isin
[0infin)
119861120572120573
119899120574(119905119898
119909) =119899119898
Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119909119898
+119898119899119898minus1
Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)
+ 120572(119899
120574minus 119898 + 1)119909
119898minus1
+120572119898 (119898 minus 1) 119899
119898minus2
Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898
minus 2) +120572 (119899120574 minus 119898 + 2)
2119909119898minus2
+ 119874 (119899minus2
)
(16)
Lemma 4 (see [10]) The polynomials119876119894119895119903120574
(119909) exist indepen-dent of 119899 and 119896 such that
119909 (1 + 120574119909)119903
119863119903
[119901119899119896120574
(119909)]
= sum
2119894+119895le119903
119894119895ge0
119899119894
(119896 minus 119899119909)119895
119876119894119895119903120574
(119909) 119901119899119896120574
(119909)
where 119863 equiv119889
119889119909
(17)
Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)
= 119874(119905120592
) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has
(119861120572120573
119899120574)(119903)
(119891 119909) =119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) 119891(119903)
(119899119905 + 120572
119899 + 120573)119889119905
(18)
Proof First
(119861120572120573
119899120574)(1)
(119891 119909)
=
infin
sum
119896=1
119901(1)
119899119896120574(119909) int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(19)
Now using the identities
119901(1)
119899119896120574(119909) = 119899 119901
119899+120574119896minus1120574(119909) minus 119901
119899+120574119896120574(119909)
119887(1)
119899119896120574(119909) = (119899 + 120574) 119887
119899+120574119896minus1120574(119909) minus 119887
119899+120574119896120574(119909)
(20)
for 119896 ge 1 we have
(119861120572120573
119899120574)(1)
(119891 119909) =
infin
sum
119896=1
119899 119901119899+120574119896minus1120574
(119909) minus 119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) = 119899119901
119899+1205740120574(119909)
sdot int
infin
0
119887119899+1205741120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) + 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896+1120574
(119905) minus 119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
International Journal of Analysis 5
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
sdot int
infin
0
(119899 + 120574) (1 + 120574119905)minus119899120574minus2
119891(119899119905 + 120572
119899 + 120573)119889119905
+ 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
(minus1
119899119887(1)
119899minus120574119896+1120574(119905)) 119891(
119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(21)
Integrating by parts we get
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
+1198992
119899 + 120573(1 + 120574119909)
minus119899120574minus1
sdot int
infin
0
(1 + 120574119905)minus119899120574minus1
119891(1)
(119899119905 + 120572
119899 + 120573)119889119905 +
119899
119899 + 120573
sdot
infin
sum
119896=1
119901119899+120574119896120574
(119909) int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(119861120572120573
119899120574)(1)
(119891 119909) =119899
119899 + 120573
infin
sum
119896=0
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
(22)
Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then
(119861120572120573
119899120574)(119894)
(119891 119909) =119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119894119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(23)
Thus using the identities (20) we have
(119861120572120573
119899120574)(119894+1)
(119891 119909)
=119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
(119899
120574+ 119894)
sdot 119901119899+120574(119894+1)119896minus1120574
(119909) minus 119901119899+120574(119894+1)119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905 minus (
119899
120574+ 119894) (1 + 120574119909)
minus119899120574minus119894minus1
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus1205741198941+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
minus119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574119894119896+119894+1120574
(119905) minus 119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)1+119894120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)119896+119894+1120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(24)
Integrating by parts we obtain
(119861120572120573
119899120574)(119894+1)
(119891 119909) =119899119894+1
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894+1
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574(119894minus1)119896+119894+1120574
(119905) 119891(119894+1)
(119899119905 + 120572
119899 + 120573)119889119905
(25)
This completes the proof of Lemma 5
3 Direct Theorems
This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Analysis
sdot 120583119899119898120574
(119909) minus119898120574 (119899 + 120573)
119899(
120572
119899 + 120573minus 119909)
2
sdot 120583119899119898minus1120574
(119909) minus 119899119909120583119899119898120574
(119909)
+(119899 + 2120574) (119899 + 120573)
119899120583119899119898+1120574
(119909)
minus (120572
119899 + 120573minus 119909)120583
119899119898120574(119909)
(13)
Hence
(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574
(119909) = 119899119909 (1 + 120574119909) 120583(1)
119899119898120574(119909)
+ 119898120583119899119898minus1120574
(119909) + 119898119899 + 1198992
119909
minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574
(119909)
+ 119898120574 (119899 + 120573) (120572
119899 + 120573minus 119909)
2
minus 119898119899(120572
119899 + 120573minus 119909)120583
119899119898minus1120574(119909)
(14)
This completes the proof of Lemma 1
Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as
119880119899119898120574
(119909) =
infin
sum
119896=0
119901119899119896120574
(119909) (119896
119899minus 119909)
119898
(15)
then 1198801198990120574
(119909) = 1 1198801198991120574
(119909) = 0 and 119899119880119899119898+1120574
(119909) = 119909(1 +
120574119909)(119880(1)
119899119898120574(119909) + 119898119880
119899119898minus1120574(119909))
Consequently for all 119909 isin [0infin) we have 119880119899119898120574
(119909) =
119874(119899minus[(119898+1)2]
)
Remark 3 It is easily verified from Lemma 1 that for each 119909 isin
[0infin)
119861120572120573
119899120574(119905119898
119909) =119899119898
Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119909119898
+119898119899119898minus1
Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)
+ 120572(119899
120574minus 119898 + 1)119909
119898minus1
+120572119898 (119898 minus 1) 119899
119898minus2
Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)
(119899 + 120573)119898
Γ (119899120574 + 1) Γ (119899120574)119899 (119898
minus 2) +120572 (119899120574 minus 119898 + 2)
2119909119898minus2
+ 119874 (119899minus2
)
(16)
Lemma 4 (see [10]) The polynomials119876119894119895119903120574
(119909) exist indepen-dent of 119899 and 119896 such that
119909 (1 + 120574119909)119903
119863119903
[119901119899119896120574
(119909)]
= sum
2119894+119895le119903
119894119895ge0
119899119894
(119896 minus 119899119909)119895
119876119894119895119903120574
(119909) 119901119899119896120574
(119909)
where 119863 equiv119889
119889119909
(17)
Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)
= 119874(119905120592
) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has
(119861120572120573
119899120574)(119903)
(119891 119909) =119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) 119891(119903)
(119899119905 + 120572
119899 + 120573)119889119905
(18)
Proof First
(119861120572120573
119899120574)(1)
(119891 119909)
=
infin
sum
119896=1
119901(1)
119899119896120574(119909) int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(19)
Now using the identities
119901(1)
119899119896120574(119909) = 119899 119901
119899+120574119896minus1120574(119909) minus 119901
119899+120574119896120574(119909)
119887(1)
119899119896120574(119909) = (119899 + 120574) 119887
119899+120574119896minus1120574(119909) minus 119887
119899+120574119896120574(119909)
(20)
for 119896 ge 1 we have
(119861120572120573
119899120574)(1)
(119891 119909) =
infin
sum
119896=1
119899 119901119899+120574119896minus1120574
(119909) minus 119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) = 119899119901
119899+1205740120574(119909)
sdot int
infin
0
119887119899+1205741120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905 minus 119899 (1 + 120574119909)
minus119899120574minus1
sdot 119891 (120572
119899 + 120573) + 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899119896+1120574
(119905) minus 119887119899119896120574
(119905) 119891(119899119905 + 120572
119899 + 120573)119889119905
International Journal of Analysis 5
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
sdot int
infin
0
(119899 + 120574) (1 + 120574119905)minus119899120574minus2
119891(119899119905 + 120572
119899 + 120573)119889119905
+ 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
(minus1
119899119887(1)
119899minus120574119896+1120574(119905)) 119891(
119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(21)
Integrating by parts we get
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
+1198992
119899 + 120573(1 + 120574119909)
minus119899120574minus1
sdot int
infin
0
(1 + 120574119905)minus119899120574minus1
119891(1)
(119899119905 + 120572
119899 + 120573)119889119905 +
119899
119899 + 120573
sdot
infin
sum
119896=1
119901119899+120574119896120574
(119909) int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(119861120572120573
119899120574)(1)
(119891 119909) =119899
119899 + 120573
infin
sum
119896=0
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
(22)
Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then
(119861120572120573
119899120574)(119894)
(119891 119909) =119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119894119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(23)
Thus using the identities (20) we have
(119861120572120573
119899120574)(119894+1)
(119891 119909)
=119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
(119899
120574+ 119894)
sdot 119901119899+120574(119894+1)119896minus1120574
(119909) minus 119901119899+120574(119894+1)119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905 minus (
119899
120574+ 119894) (1 + 120574119909)
minus119899120574minus119894minus1
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus1205741198941+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
minus119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574119894119896+119894+1120574
(119905) minus 119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)1+119894120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)119896+119894+1120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(24)
Integrating by parts we obtain
(119861120572120573
119899120574)(119894+1)
(119891 119909) =119899119894+1
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894+1
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574(119894minus1)119896+119894+1120574
(119905) 119891(119894+1)
(119899119905 + 120572
119899 + 120573)119889119905
(25)
This completes the proof of Lemma 5
3 Direct Theorems
This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 5
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
sdot int
infin
0
(119899 + 120574) (1 + 120574119905)minus119899120574minus2
119891(119899119905 + 120572
119899 + 120573)119889119905
+ 119899
infin
sum
119896=1
119901119899+120574119896120574
(119909)
sdot int
infin
0
(minus1
119899119887(1)
119899minus120574119896+1120574(119905)) 119891(
119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(21)
Integrating by parts we get
(119861120572120573
119899120574)(1)
(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
+1198992
119899 + 120573(1 + 120574119909)
minus119899120574minus1
sdot int
infin
0
(1 + 120574119905)minus119899120574minus1
119891(1)
(119899119905 + 120572
119899 + 120573)119889119905 +
119899
119899 + 120573
sdot
infin
sum
119896=1
119901119899+120574119896120574
(119909) int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
minus 119899 (1 + 120574119909)minus119899120574minus1
119891(120572
119899 + 120573)
(119861120572120573
119899120574)(1)
(119891 119909) =119899
119899 + 120573
infin
sum
119896=0
119901119899+120574119896120574
(119909)
sdot int
infin
0
119887119899minus120574119896+1120574
(119905) 119891(1)
(119899119905 + 120572
119899 + 120573)119889119905
(22)
Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then
(119861120572120573
119899120574)(119894)
(119891 119909) =119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574119894119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(23)
Thus using the identities (20) we have
(119861120572120573
119899120574)(119894+1)
(119891 119909)
=119899119894
Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
(119899
120574+ 119894)
sdot 119901119899+120574(119894+1)119896minus1120574
(119909) minus 119901119899+120574(119894+1)119896120574
(119909) int
infin
0
119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905 minus (
119899
120574+ 119894) (1 + 120574119909)
minus119899120574minus119894minus1
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus1205741198941+119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
minus119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
119887119899minus120574119894119894120574
(119905) 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574119894119896+119894+1120574
(119905) minus 119887119899minus120574119894119896+119894120574
(119905)
sdot 119891(119894)
(119899119905 + 120572
119899 + 120573)119889119905
=119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
119901119899+120574(119894+1)0120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)1+119894120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
+119899119894
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=1
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
(minus1
119899120574 minus 119894119887(1)
119899minus120574(119894minus1)119896+119894+1120574(119905))119891
(119894)
(119899119905 + 120572
119899 + 120573)119889119905
(24)
Integrating by parts we obtain
(119861120572120573
119899120574)(119894+1)
(119891 119909) =119899119894+1
Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)
(119899 + 120573)119894+1
Γ (119899120574 + 1) Γ (119899120574)
sdot
infin
sum
119896=0
119901119899+120574(119894+1)119896120574
(119909)
sdot int
infin
0
119887119899minus120574(119894minus1)119896+119894+1120574
(119905) 119891(119894+1)
(119899119905 + 120572
119899 + 120573)119889119905
(25)
This completes the proof of Lemma 5
3 Direct Theorems
This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
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Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Analysis
We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le
119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot
120583on the
class 119862120583[0infin) is defined as 119891
120583= sup
0le119905ltinfin|119891(119905)|119905
minus120583
It canbe easily verified that the operators 119861120572120573
119899120574(119891 119909) are well defined
for 119891 isin 119862120583[0infin)
Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point
119909 isin (0infin) Then one has
lim119899rarrinfin
(119861120572120573
119899120574)(119903)
(119891 119909) = 119891(119903)
(119909) (26)
Proof By Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903
(27)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence
(119861120572120573
119899120574)(119903)
(119891 119909) =
119903
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909)
+ (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903
119909)
= 1198771+ 1198772
(28)
First to estimate 1198771 using binomial expansion of ((119899119905 +
120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have
1198771=
119903
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
=119891(119903)
(119909)
119903119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
= 119891(119903)
(119909) 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
997888rarr 119891(119903)
(119909) as 119899 997888rarr infin
(29)
Next applying Lemma 4 we obtain
1198772= int
infin
0
119882(119903)
119899120574(119905 119909) 120598 (119905 119909) (
119899119905 + 120572
119899 + 120573minus 119909)
119903
119889119905
100381610038161003816100381611987721003816100381610038161003816 le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) |120598 (119905 119909)|
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+Γ (119899120574 + 119903 + 2)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
|120598 (0 119909)|
sdot
10038161003816100381610038161003816100381610038161003816
120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
(30)
The second term in the above expression tends to zero as 119899 rarr
infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872
3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)
119903
| le
1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|
120591 for |119905 minus 119909| ge 120575 Therefore
100381610038161003816100381611987721003816100381610038161003816 le 119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=0
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot 120576 int|119905minus119909|lt120575
119887119899119896120574
(119909)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
119903
119889119905
+ int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 = 1198773+ 1198774
(31)
Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain
100381610038161003816100381611987731003816100381610038161003816 le 120576119872
3sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2119903
119889119905
12
(32)
Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899
1199032
)119874(119899minus1199032
)
= 120576 sdot 119874(1)
Again using the Cauchy-Schwarz inequality and Lemma1 we get
100381610038161003816100381611987741003816100381610038161003816 le 119872
4sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909)
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905)
10038161003816100381610038161003816100381610038161003816
119899119905 + 120572
119899 + 120573minus 119909
10038161003816100381610038161003816100381610038161003816
120591
119889119905 le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
|119896 minus 119899119909|119895
119901119899119896120574
(119909) int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905
12
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
le 1198724
sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
(119896 minus 119899119909)2119895
119901119899119896120574
(119909)
12
sdot
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
2120591
119889119905
12
= sum
2119894+119895le119903
119894119895ge0
119899119894
119874(1198991198952
)119874 (119899minus1205912
) = 119874 (119899(119903minus120591)2
) = 119900 (1)
(33)
Collecting the estimation of1198771ndash1198774 we get the required result
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
International Journal of Analysis 7
Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin
(0infin) then
lim119899rarrinfin
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)
(119909)
+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)
(119909)
+ 119909 (1 + 120574119909) 119891(119903+2)
(119909)
(34)
Proof Using Taylorrsquos expansion of 119891 we have
119891 (119905) =
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119905 minus 119909)
119894
+ 120598 (119905 119909) (119905 minus 119909)119903+2
(35)
where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583
) 119905 rarr
infin for 120583 gt 0Applying Lemma 1 we have
119899 (119861120572120573
119899120574)(119903)
(119891 119909) minus 119891(119903)
(119909)
= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894(119861120572120573
119899120574)(119903)
((119905 minus 119909)119894
119909) minus 119891(119903)
(119909)
+ 119899 (119861120572120573
119899120574)(119903)
(120598 (119905 119909) (119905 minus 119909)119903+2
119909)
= 1198641+ 1198642
(36)
First we have
1198641= 119899
119903+2
sum
119894=0
119891(119894)
(119909)
119894
119894
sum
119895=0
(119894
119895) (minus119909)
119894minus119895
(119861120572120573
119899120574)(119903)
(119905119895
119909)
minus 119899119891(119903)
(119909) =119891(119903)
(119909)
119903119899 (119861
120572120573
119899120574)(119903)
(119905119903
119909) minus 119903
+119891(119903+1)
(119909)
(119903 + 1)119899 (119903 + 1) (minus119909) (119861
120572120573
119899120574)(119903)
(119905119903
119909)
+ (119861120572120573
119899120574)(119903)
(119905119903+1
119909) +119891(119903+2)
(119909)
(119903 + 2)
sdot 119899 (119903 + 2) (119903 + 1)
21199092
(119861120572120573
119899120574)(119903)
(119905119903
119909) + (119903 + 2)
sdot (minus119909) (119861120572120573
119899120574)(119903)
(119905119903+1
119909) + (119861120572120573
119899120574)(119903)
(119905119903+2
119909)
= 119891(119903)
(119909) 119899 119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)minus 1
+119891(119903+1)
(119909)
(119903 + 1)119899(119903 + 1) (minus119909)
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903
+119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903 + 120572(119899
120574
minus 119903) 119903 +119891(119903+2)
(119909)
(119903 + 2)119899(
(119903 + 1) (119903 + 2)
21199092
sdot119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)
sdot 119899119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 1)119909
+(119903 + 1) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903+1
Γ (119899120574 + 1) Γ (119899120574)
119899119903
+ 120572(119899
120574minus 119903) 119903
+119899119903+2
Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
(119903 + 2)
21199092
+(119903 + 2) 119899
119903+1
Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)
(119899 + 120573)119903+2
Γ (119899120574 + 1) Γ (119899120574)
119899 (119903
+ 1) + 120572(119899
120574minus 119903 minus 1) (119903 + 1)119909
+120572 (119903 + 1) (119903 + 2) 119899
119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)119899119903
+120572 (119899120574 minus 119903)
2 119903)
(37)
Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)
(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864
2rarr 0
as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4
Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem
Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575
[119886 119887]) for a function continuous on [119886 119887] is defined by
120596119898(119891 120575 [119886 119887])
= sup 1003816100381610038161003816Δ119898
ℎ119891 (119909)
1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]
(38)
For119898 = 1 120596119898(119891 120575) is usual modulus of continuity
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Analysis
Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt
1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le 11987211205962(119891(119903)
119899minus12
[1198861 1198871]) + 119872
2119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(39)
where1198721= 1198721(119903) and119872
2= 1198722(119903 119891)
Proof Let us assume that 0 lt 119886 lt 1198861
lt 1198871
lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891
1205782
corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886
1 1198871] as follows
1198911205782
(119905) = 120578minus2
∬
1205782
minus1205782
(119891 (119905) minus Δ2
ℎ119891 (119905)) 119889119905
11198891199052 (40)
where ℎ = (1199051+ 1199052)2 and Δ
2
ℎis the second order forward
difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891
1205782are known as the Steklov mean of order 2
which satisfy the following properties [11]
(a) 1198911205782
has continuous derivatives up to order 2 over[1198861 1198871]
(b) 119891(119903)1205782
119862[11988611198871]le 1198721120578minus119903
1205962(119891 120578 [119886 119887]) 119903 = 1 2
(c) 119891 minus 1198911205782
119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])
(d) 1198911205782
119862[11988611198871]le 1198723119891120583
where119872119894 119894 = 1 2 3 are certain constants which are different
in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573
119899120574
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 sdot) minus 119891(119903)
1003817100381710038171003817100381710038171003817119862[11988611198871]
le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(119891 minus 1198911205782
sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817((119861120572120573
119899120574)(119903)
1198911205782
sdot) minus 119891(119903)
1205782
1003817100381710038171003817100381710038171003817119862[11988611198871]
+10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[11988611198871]
= 1198751+ 1198752+ 1198753
(41)
Since 119891(119903)
1205782= (119891(119903)
)1205782
(119905) by property (c) of the function 1198911205782
we get
1198753le 11987241205962(119891(119903)
120578 [119886 119887]) (42)
Next on an application of Theorem 7 it follows that
1198752le 1198725119899minus1
119903+2
sum
119894=119903
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887] (43)
Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that
10038171003817100381710038171003817119891(119894)
1205782
10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782
10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)
1205782
10038171003817100381710038171003817119862[119886119887] (44)
Therefore by applying properties (c) and (d) of the function1198911205782 we obtain
1198752le 1198727sdot 119899minus1
1003817100381710038171003817119891
1003817100381710038171003817120583 + 120575minus2
1205962(119891(119903)
120583 [119886 119887]) (45)
Finally we will estimate 1198751 choosing 119886
lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886
lowast
lt 1198861lt 1198871lt 119887lowast
lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then
1198751le
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
+
1003817100381710038171003817100381710038171003817(119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) sdot)
1003817100381710038171003817100381710038171003817119862[11988611198871]
= 1198754+ 1198755
(46)
By Lemma 5 we have
(119861120572120573
119899120574)(119903)
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) 119909)
=119899119903
Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)
(119899 + 120573)119903
Γ (119899120574 + 1) Γ (119899120574)
infin
sum
119896=0
119901119899+120574119903119896120574
(119909)
sdot int
infin
0
119887119899minus120574119903119896+119903120574
(119905) ℏ (119905)
sdot (119891(119903)
(119899119905 + 120572
119899 + 120573) minus 119891
(119903)
1205782(119899119905 + 120572
119899 + 120573))119889119905
(47)
Hence100381710038171003817100381710038171003817(119861120572120573
119899120574)119903
(ℏ (119905) (119891 (119905) minus 1198911205782
(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]
le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]
(48)
Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886
lowast
119887lowast
] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575
Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have
119868 equiv (119861120572120573
119899120574)(119903)
((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782
(119905)) 119909)
|119868| le sum
2119894+119895le119903
119894119895ge0
119899119894
10038161003816100381610038161003816119876119894119895119903120574
(119909)10038161003816100381610038161003816
119909 (1 + 120574119909)119903
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int
infin
0
119887119899119896120574
(119905) (1 minus ℏ (119905))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
119899119905 + 120572
119899 + 120573) minus 1198911205782
(119899119905 + 120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816119889119905
+Γ (119899120574 + 119903)
Γ (119899120574)(1 + 120574119909)
minus119899120574minus119903
(1 minus ℏ (0))
sdot
10038161003816100381610038161003816100381610038161003816119891 (
120572
119899 + 120573) minus 1198911205782
(120572
119899 + 120573)
10038161003816100381610038161003816100381610038161003816
(49)
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 9
For sufficiently large 119899 the second term tends to zero Thus
|119868| le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 sum
2119894+119895le119903
119894119895ge0
119899119894
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
sdot int|119905minus119909|ge120575
119887119899119896120574
(119905) 119889119905 le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|119895
(int
infin
0
119887119899119896120574
(119905) 119889119905)
12
sdot (int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
le 1198729
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
sum
2119894+119895le119903
119894119895ge0
119899119894
sdot (
infin
sum
119896=1
119901119899119896120574
(119909) |119896 minus 119899119909|2119895
)
12
(
infin
sum
119896=1
119901119899119896120574
(119909)
sdot int
infin
0
119887119899119896120574
(119905) (119899119905 + 120572
119899 + 120573minus 119909)
4119898
119889119905)
12
(50)
Hence by using Remark 2 and Lemma 1 we have
|119868| le 11987210
10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898
119874(119899(119894+(1198952)minus119898)
) le 11987211119899minus119902 1003817100381710038171003817119891
1003817100381710038171003817120583 (51)
where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872
11119899minus1
119891120583 Therefore by property (c) of the
function 1198911205782
(119905) we get
1198751le 1198728
10038171003817100381710038171003817119891(119903)
minus 119891(119903)
1205782
10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
le 119872121205962(119891(119903)
120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891
1003817100381710038171003817120583
(52)
Choosing 120578 = 119899minus12 the theorem follows
Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]
A Durrmeyer type 119902-analogue of the 119861120572120573
119899120574(119891 119909) is intro-
duced as follows
119861120572120573
119899120574119902(119891 119909)
=
infin
sum
119896=1
119901119902
119899119896120574(119909) int
infin119860
0
119902minus119896
119887119902
119899119896120574(119905) 119891(
[119899]119902119905 + 120572
[119899]119902+ 120573
)119889119902119905
+ 119901119902
1198990120574(119909) 119891(
120572
[119899]119902+ 120573
)
(53)
where
119901119902
119899119896120574(119909) = 119902
11989622
Γ119902(119899120574 + 119896)
Γ119902(119896 + 1) Γ
119902(119899120574)
sdot(119902120574119909)
119896
(1 + 119902120574119909)(119899120574)+119896
119902
119887119902
119899119896120574(119909) = 120574119902
11989622
Γ119902(119899120574 + 119896 + 1)
Γ119902(119896) Γ119902(119899120574 + 1)
sdot(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119902
int
infin119860
0
119891 (119909) 119889119902119909 = (1 minus 119902)
infin
sum
119899=minusinfin
119891(119902119899
119860)
119902119899
119860 119860 gt 0
(54)
Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this research article
Acknowledgments
The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article
References
[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012
[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012
[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013
[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009
[5] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 4 article 125 2006
[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Analysis
[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015
[8] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[9] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005
[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969
[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971
[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008
[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997
[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006
[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014
[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of