Research Article Some Definition of Hartley-Hilbert and...
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Research Article Some Definition of Hartley-Hilbert and Fourier-Hilbert Transforms in a Quotient Space of Boehmians S. K. Q. Al-Omari Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa’ Applied University, Amman 11134, Jordan Correspondence should be addressed to S. K. Q. Al-Omari; [email protected] Received 20 May 2014; Accepted 16 August 2014; Published 6 November 2014 Academic Editor: Tepper L Gill Copyright © 2014 S. K. Q. Al-Omari. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the Hartley-Hilbert and Fourier-Hilbert transforms on a quotient space of Boehmians. e investigated transforms are well-defined and linear mappings in the space of Boehmians. Further properties are also obtained. 1. Introduction e Hilbert transform of a function () via the Hartley transform is defined in [1, 2] as () hh () = hh () () = 1 ∫ ∞ 0 (() o h () cos () + () e h () sin ()) , (1) where () e h and () o h are, respectively, the even and odd components of the Hartley transform () h given as [3] () h () = ∫ ∞ −∞ () cas () := () e h () + () o h () , (2) where cas() = cos() + sin(). Let () be a casual function; that is, () = 0, for >0, and then () o h () and () e h () are related by a Hilbert transform pair as [3] () o h () = 1 ∫ ∞ −∞ () e h () − , () e h () = 1 ∫ ∞ −∞ () o h () − . (3) e Hilbert transform of () via the Fourier transform is defined by () fh () = fh () () = 1 ∫ ∞ 0 (() i f () cos () − () r f () sin ()) , (4) where () r f () and () i f () are, respectively, the real and imaginary components of the Fourier transform given as () f () = ∫ ∞ −∞ () − := () r f () − () i f () . (5) e Hartley transform is extended to Boehmians in [4] and to strong Boehmians in [5]. e Hartley-Hilbert and Fourier-Hilbert transforms were discussed in various spaces of distributions and spaces of Boehmians in [1, 6]. In this paper, aim at investigating the Hartley-Hilbert transform on the context of Boehmians. Investigating the later transform is analogous. 2. Spaces of Quotients (Spaces of Boehmians) One of the most youngest generalizations of functions and more particularly of distributions is the theory of Boehmians. e idea of the construction of Boehmians was initiated by the concept of regular operators [7]. Regular operators form Hindawi Publishing Corporation Journal of Mathematics Volume 2014, Article ID 790161, 5 pages http://dx.doi.org/10.1155/2014/790161
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Research ArticleSome Definition of Hartley-Hilbert and Fourier-HilbertTransforms in a Quotient Space of Boehmians
S K Q Al-Omari
Department of Applied Sciences Faculty of Engineering Technology Al-Balqarsquo Applied University Amman 11134 Jordan
Correspondence should be addressed to S K Q Al-Omari skqalomarifetedujo
Received 20 May 2014 Accepted 16 August 2014 Published 6 November 2014
Academic Editor Tepper L Gill
Copyright copy 2014 S K Q Al-Omari This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We investigate the Hartley-Hilbert and Fourier-Hilbert transforms on a quotient space of Boehmians The investigated transformsare well-defined and linear mappings in the space of Boehmians Further properties are also obtained
1 Introduction
The Hilbert transform of a function 119891(119910) via the Hartleytransform is defined in [1 2] as
(119891)hh
(119910)
= hh (119891) (119910)
=1
120587intinfin
0
((119891)o
h(119909) cos (119909119910) + (119891)
e
h(119909) sin (119909119910)) 119889119909
(1)
where (119891)eh and (119891)oh are respectively the even and oddcomponents of the Hartley transform (119891)h given as [3]
(119891)h(119909) = int
infin
minusinfin
119891 (119910) cas (119909119910) 119889119910 = (119891)e
h(119909) + (119891)
o
h(119909)
(2)
where cas(119909119910) = cos(119909119910) + sin(119909119910)Let 119891(119910) be a casual function that is 119891(119910) = 0 for
119910 gt 0 and then (119891)oh(119909) and (119891)eh(119909) are related by a Hilberttransform pair as [3]
(119891)o
h(119909) =
1
120587intinfin
minusinfin
(119891)e
h(119910)
119909 minus 119910119889119910
(119891)e
h(119909) =
1
120587intinfin
minusinfin
(119891)o
h(119910)
119909 minus 119910119889119910
(3)
The Hilbert transform of 119891(119910) via the Fourier transformis defined by
(119891)fh
(119910)
= fh (119891) (119910)
=1
120587intinfin
0
((119891)i
f(119909) cos (119909119910) minus (119891)
r
f(119909) sin (119909119910)) 119889119909
(4)
where (119891)rf (119909) and (119891)if(119909) are respectively the real andimaginary components of the Fourier transform given as
(119891)f(119909) = int
infin
minusinfin
119891 (119910) 119890minus119894119909119910119889119910 = (119891)r
f(119909) minus 119894 (119891)
i
f(119909) (5)
The Hartley transform is extended to Boehmians in [4]and to strong Boehmians in [5] The Hartley-Hilbert andFourier-Hilbert transforms were discussed in various spacesof distributions and spaces of Boehmians in [1 6]
In this paper 119894 aim at investigating the Hartley-Hilberttransform on the context of Boehmians Investigating thelater transform is analogous
2 Spaces of Quotients (Spaces of Boehmians)
One of the most youngest generalizations of functions andmore particularly of distributions is the theory of BoehmiansThe idea of the construction of Boehmians was initiated bythe concept of regular operators [7] Regular operators form
Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 790161 5 pageshttpdxdoiorg1011552014790161
2 Journal of Mathematics
a subalgebra of the field of Mikusinski operators and theyinclude only such functions whose support is bounded fromthe left In a concrete case the space of Boehmians contains allregular operators all distributions and some objects whichare neither operators nor distributions
The construction of Boehmians is similar to the construc-tion of the field of quotients and in some cases it gives justthe field of quotients On the other hand the constructionis possible where there are zero divisors such as space C(the space of continuous functions) with the operations ofpointwise additions and convolution
A number of integral transforms have been extended toBoehmian spaces in the recent past by many authors suchas Roopkumar in [8 9] Mikusinski and Zayed in [10] Al-Omari and Kilicman in [1 6 11] Karunakaran and Vembu[12] Karunakaran and Roopkumar [13] Al-Omari et al in[14] and many others
For abstract construction of Boehmian spaces we refer to[14ndash16]
By ⊛ denote the Mellin-type convolution product of firstkind defined by [8]
(120601 ⊛ 120595) (119905) = intinfin
0
119911minus1120601 (119905119911minus1) 119892 (119911) 119889119911 (6)
Properties of ⊛ are presented as follows [8]
(i) (120601 ⊛ 120595) (119905) = (120595 ⊛ 120601) (119905)(ii) ((120601 + 120595) ⊛ 120593) (119905) = (120601 ⊛ 120593) (119905) + (120595 ⊛ 120593) (119905)(iii) (120572120601 ⊛ 120595) (119905) = 120572 (120595 ⊛ 120601) (119905) 120572 is complex number(iv) ((120601 ⊛ 120595) ⊛ 120593) (119905) = (120601 ⊛ (120595 ⊛ 120593)) (119905)
By lowast we denote the convolution product defined by
(120601 lowast 120595) (119910) = intinfin
0
119911minus1120601 (119910119911minus1) 120595 (119911) 119889119911 (7)
By 120599(R+) denote the space of test functions of bounded
supports defined on R+that vanish more rapidly than every
power of 119909 as 119909 rarr infin Denote by 1205991the subset of 120599(R
+)
defined by
1205991= 120593 isin 120599 (R
+) intinfin
0
120593 (119909) 119889119909 = 1 (8)
Let Δ be the set of delta sequences satisfying the followingproperties
(1) 120575119899 isin 120599(R
+)
(2) intinfin0
120575119899(119909) 119889119909 = 1 119899 isin N
(3) intinfin0
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 lt infin 119899 isin N
(4) supp 120575119899(119909) sube [119886 119887
119899] 0 lt 119886
119899lt 119887119899and 119886119899 119887119899
rarr 0 as119899 rarr infin where
supp 120575119899(119909) = 119909 isin R
+ 120575119899(119909) = 0 forall119899 isin N (9)
Now we establish the following theorem
Theorem 1 Let 120601 120595 isin 120599(R+) then one has (120601 ⊛ 120595)
hh(119910) =
(120601hh lowast 120595) (119910)
Proof Let 120601 and 120595 be in 120599(R+) then we have
(120601 ⊛ 120595)hh
(119910) = intinfin
0
((120601 ⊛ 120595)o
h(119909) cos (119909119910)
+ (120601 ⊛ 120595)e
h(119909) sin (119909119910)) 119889119909
ie = intinfin
0
(intinfin
0
(120601 ⊛ 120595) (119905) sin (119909119905) 119889119905 cos (119909119910)
+ intinfin
0
(120601 ⊛ 120595) (119905) cos (119909119905) 119889119905
times sin (119909119910)) 119889119909
ie = intinfin
0
(intinfin
0
(intinfin
0
(120601 (119905119911minus1) sin (119909119905) cos (119909119910)
+ 120601 (119905119911minus1) cos (119909119905)
times sin (119909119910)) 119889119905) 119889119909)
times 120595 (119911) 119911minus1119889119911
(10)
By the change of variables 119905119911minus1 = 119908 we get that
(120601 ⊛ 120595)hh
(119910) = intinfin
0
(intinfin
0
(intinfin
0
(120601 (119908) sin (119909119911119908) cos (119909119910)
+ 120601 (119908) cos (119909119911119908)
times sin (119909119910)) 119889119908)119889119909)
times 120595 (119911) 119911 119889119911
ie = intinfin
0
(intinfin
0
(120601oh (119909119911) cos (119909119910)
+120601eh (119909119911) sin (119909119910)) 119889119909)
times 120595 (119911) 119911 119889119911
(11)
Once again by the change of variables 119909119911 = 119903 we get
(120601 ⊛ 120595)hh
(119910) = intinfin
0
119911minus1120601hh (119910119911minus1) 120595 (119911) 119889119911 (12)
This completes the proof of the theorem
Now we establish the following theorem
Theorem 2 Let 120601 120595 and 120579 be test functions in 120599(R+) then
120601 lowast (120595 ⊛ 120579) = (120601 lowast 120595) lowast 120579
Journal of Mathematics 3
Proof Let 119910 gt 0 be given Then from (6) and (7) we have
(120601 lowast (120595 ⊛ 120579)) (119910) = intinfin
0
119911minus1120601 (119910119911minus1) (120595 ⊛ 120579) (119911) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (intinfin
0
119905minus1120595 (119911119905minus1) 120579 (119905) 119889119905) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (120595 lowast 120579) (119911) 119889119911
(13)
Hence the theorem is completely proved
Theorem 3 Let 120593 120595 isin 1205991 then 120593 ⊛ 120595 = 120595 ⊛ 120593
Proof Proof of this theorem follows from technique similarto that of Theorem 2 Therefore we omit the details
Theorem 4 Let 1206011 1206012
isin 120599(R+) and 120593
1 1205932
isin 1205991 then the
following hold
(1) (1206011+ 1206012) lowast 1205931= 1206011lowast 1205931+ 1206012lowast 1205931
(2) if 120601119899
rarr 120601 isin 120599(R+) as 119899 rarr infin then 120601
119899lowast120593 rarr 120601lowast120593
as 119899 rarr infin
(3) 1206011lowast (1205931⊛ 1205932) = (120601
1lowast 1205931) lowast 1205932
(4) 1206011lowast 1205931isin 120599(R
+)
Proof Proof of (1) and (2) follows from elementary integralcalculus Proof of (3) is already obtained inTheorem 2 Proofof (4) is obvious by the properties of the product (7) Furtherdetails are thus avoided
This completes the proof of the theorem
Theorem 5 Let 120601 isin 120599(R+) and 120575
119899 isin Δ then 120601 lowast 120575
119899rarr 120601
as 119899 rarr infin
Proof Let 119870 be a compact set containing the support of 120601then by Property 2 of delta sequences we have
10038161003816100381610038161003816D119896
119910(120601 lowast 120575
119899minus 120601) (119910)
10038161003816100381610038161003816
le int119870
10038161003816100381610038161003816D119896
119910(119911minus1120601 (119910119911minus1) minus 120601 (119910))
100381610038161003816100381610038161003816100381610038161003816120575119899 (119910)
1003816100381610038161003816 119889119911(14)
Hence (14) goes to zero as 119899 rarr infinHence the theorem is completely proved
The Boehmian spaceB (120599 (1205991 ⊛) lowast Δ) is constructed
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899lowast 120595119899+ 119892119899lowast 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(15)
where 120572 isin C the field of complex numbers
The operation lowast and the differentiation are defined by
[119891119899
120596119899] lowast [
119892119899
120595119899] = [
119891119899lowast 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(16)
Similarly the spaceB (120599 (1205991 ⊛) ⊛ Δ) can be proved
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899⊛ 120595119899+ 119892119899⊛ 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(17)
The operation lowast and the differentiation are defined by
[119891119899
120596119899] ⊛ [
119892119899
120595119899] = [
119891119899⊛ 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(18)
As next let 120601 isin 120599(R+) then (120601)
hh(119910) isin 120599(R
+)
For some details we by definitions have that10038161003816100381610038161003816D119896
119910(120601)
hh(119910)
10038161003816100381610038161003816
le1
120587intinfin
0
D119896
119910(10038161003816100381610038161003816((120601)
o
h(119909) cos (119909119910) + (120601)
e
h(119909) sin (119909119910))
10038161003816100381610038161003816) 119889119909
le1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 +1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909
(19)
But since1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 =1
120587intinfin
0
int119887
119886
10038161003816100381610038161003816119909119896100381610038161003816100381610038161003816100381610038161003816120601 (119910)
1003816100381610038161003816 119889119910 119889119909
997888rarr 0
(20)
for every power of 119909 as 119909 rarr infin where [119886 119887] is a compact setcontaining the support of 120601 similarly we see that
1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909 997888rarr 0 (21)
for every power of 119909 as 119909 rarr infin Therefore we get that(120601)
hh(119910) isin 120599(R
+)
3 Hartley-Hilbert Transform of Quotients
Let [120601119899 120575119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) then we define the
extended Hartley-Hilbert transform of [120601119899 120575119899] as
([120601119899
120575119899])
hh
= [
[
(120601119899)hh
120575119899
]
]
(22)
in the quotient spaceB (120599 (1205991 ⊛) lowast Δ)
4 Journal of Mathematics
Theorem 6 The operator ([120601119899 120575119899])
hhis well defined and
linear fromB (120599 (1205991 ⊛) ⊛ Δ) intoB (120599 (120599
1 ⊛) lowast Δ)
Proof We show that ([120601119899 120575119899])
hhis well defined
Let [120595119899 120583119899] = [120585
119899 120598119899] in B (120599 (120599
1 ⊛) ⊛ Δ)
then by the concept of quotients of the spaceB (120599 (120599
1 ⊛) ⊛ Δ) we have
120595119899⊛ 120598119898
= 120585119898
⊛ 120583119899 (23)
Hence by Theorem 1 we get from (23) that
(120595119899)hhlowast 120598119898
= (120585119898)hhlowast 120583119899 (24)
Hence from (24) it follows that
(120595119899)hh
120583119899
sim(120585119899)hh
120598119899
(25)
inB (120599 (1205991 ⊛) lowast Δ)
Therefore
[
[
(120595119899)hh
120583119899
]
]
= [
[
(120585119899)hh
120598119899
]
]
(26)
inB (120599 (1205991 ⊛) lowast Δ)
That is
([120595119899
120583119899])
hh
= ([120585119899
120598119899])
hh
(27)
To show that ([120601119899 120575119899])
hhis linear let [119891
119899 120598119899]
[119892119899 120591119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) 120581 120578 isin R then
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= ([120581 119891119899
120598119899
] + [120578 119892119899
120591119899
])hh
ie = ([(120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899
120598119899⊛ 120591119899
])hh
ie = [
[
((120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899)hh
120598119899⊛ 120591119899
]
]
(28)
Linearity of the classical Hartley-Hilbert transformimplies
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899⊛ 120591119899)hh
+ 120578 (119892119899⊛ 120598119899)hh
120598119899 ⊛ 120591119899
]
]
(29)
Theorem 1 gives
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
lowast 120591119899 + 120578 (119892
119899)hh
lowast 120598119899
120598119899 ⊛ 120591119899
]
]
(30)
Thus by addition of Boehmians
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
120598119899
]
]
+ [
[
120578 (119892119899)hh
120591119899
]
]
(31)
Hence
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= 120581[
[
(119891119899)hh
120598119899
]
]
+ 120578[
[
(119892119899)hh
120591119899
]
]
(32)
The proof of the theorem is completed
Theorem 7 The necessary and sufficient condition for[119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) to be in the range of
([119892119899 120595119899])
hhis that 119892
119899belongs to range of the classical hh
for every 119899 isin N
Proof Let [119892119899 120595119899] be in the range of ([119892
119899 120595119899])
hh
then of course 119892119899belongs to the range of the classical hh forall119899 isin
NTo establish the converse let119892
119899be in the range of hhforall119899 isin
N Then there is 120601119899isin 120599(R
+) such that hh120601
119899= 119892119899 119899 isin N
Since [119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) we get 119892
119899lowast
120595119898
= 119892119898lowast 120595119899 forall119898 119899 isin N
Therefore Theorem 1 yields
hh (120601119899⊛ 120595119898) = hh (120601
119898⊛ 120595119899) forall119898 119899 isin N (33)
where 120601119899isin 120599(R
+) and 120595
119899 isin Δ forall119899 isin N
Thus 120601119899⊛ 120595119898
= 120601119898
⊛ 120595119899 119898 119899 isin N
Hence
[120601119899
120595119899] isin B (120599 (120599
1 ⊛) ⊛ Δ)
([120601119899
120595119899])
hh
= [119892119899
120595119899]
(34)
The theorem is therefore completely proved
Theorem 8 The transform ([120601119899 120595119899])
hhis consistent with
hh 120599(R+) rarr 120599(R
+)
Journal of Mathematics 5
Proof For every 120601 isin 120599(R+) let 120573 isin B (120599 (120599
1 ⊛) ⊛ Δ) be its
representative then we have 120573 = [120601 ⊛ 120595119899 120595119899] forall119899 isin N
120595119899 isin Δ For all 119899 isin N it is clear that 120595
119899 is independent
from the representativeByTheorem 1 we have
(120573)ℎℎ
= ([120601 ⊛ 120595
119899
120595119899
])hh
= [hh (120601 ⊛ 120595
119899)
120595119899
] = [hh120601 lowast 120595
119899
120595119899
]
(35)
which is the representative of hh120601 in the space 120599(R+)
Hence the proof of this theorem is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author expresses his gratitude to the referees for valuablesuggestions
References
[1] S K Q Al-Omari and A Kılıcman ldquoSome remarks on theextended Hartley-Hilbert and Fourier-Hilbert transforms ofboehmiansrdquo Abstract and Applied Analysis vol 2013 Article ID348701 6 pages 2013
[2] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[3] R P Millane ldquoAnalytical properties of the Hartley transformand their implicationsrdquo Proceedings of the IEEE vol 82 no 3pp 413ndash428 1994
[4] S K Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[5] S K Al-Omari ldquoHartley transforms on a certain space ofgeneralized functionsrdquo Georgian Mathematical Journal vol 20no 3 pp 415ndash426 2013
[6] S K Q Al-Omari and A Kilicman ldquoOn the generalizedHartley-Hilbert and Fourier-Hilbert transformsrdquo Advances inDifference Equations vol 2012 article 232 2012
[7] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[8] R Roopkumar ldquoMellin transform for Boehmiansrdquo Bulletin ofthe Institute of Mathematics Academia Sinica New Series vol4 no 1 pp 75ndash96 2009
[9] R Roopkumar ldquoGeneralized Radon transformrdquo Rocky Moun-tain Journal of Mathematics vol 36 no 4 pp 1375ndash1390 2006
[10] P Mikusinski and A Zayed ldquoThe Radon transform of Boehmi-ansrdquo Proceedings of the American Mathematical Society vol 118no 2 pp 561ndash570 1993
[11] S K Al-Omari and A Kilicman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2021 Article ID 405368 14 pages2012
[12] V Karunakaran and R Vembu ldquoHilbert transform on periodicboehmiansrdquo Houston Journal of Mathematics vol 29 no 2 pp437ndash454 2003
[13] V Karunakaran and R Roopkumar ldquoOperational calculus andFourier transform on Boehmiansrdquo Colloquium Mathematicumvol 102 no 1 pp 21ndash32 2005
[14] S K Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 6 pp 453ndash4622008
[15] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[16] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science AmsterdamThe Netherlands 1997
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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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 2: Research Article Some Definition of Hartley-Hilbert and ...downloads.hindawi.com/journals/jmath/2014/790161.pdf · andtostrongBoehmiansin[]. e Hartley-Hilbert and Fourier-Hilbert](https://reader035.fdocuments.in/reader035/viewer/2022062602/5edc9aa7ad6a402d666756a2/html5/thumbnails/2.jpg)
2 Journal of Mathematics
a subalgebra of the field of Mikusinski operators and theyinclude only such functions whose support is bounded fromthe left In a concrete case the space of Boehmians contains allregular operators all distributions and some objects whichare neither operators nor distributions
The construction of Boehmians is similar to the construc-tion of the field of quotients and in some cases it gives justthe field of quotients On the other hand the constructionis possible where there are zero divisors such as space C(the space of continuous functions) with the operations ofpointwise additions and convolution
A number of integral transforms have been extended toBoehmian spaces in the recent past by many authors suchas Roopkumar in [8 9] Mikusinski and Zayed in [10] Al-Omari and Kilicman in [1 6 11] Karunakaran and Vembu[12] Karunakaran and Roopkumar [13] Al-Omari et al in[14] and many others
For abstract construction of Boehmian spaces we refer to[14ndash16]
By ⊛ denote the Mellin-type convolution product of firstkind defined by [8]
(120601 ⊛ 120595) (119905) = intinfin
0
119911minus1120601 (119905119911minus1) 119892 (119911) 119889119911 (6)
Properties of ⊛ are presented as follows [8]
(i) (120601 ⊛ 120595) (119905) = (120595 ⊛ 120601) (119905)(ii) ((120601 + 120595) ⊛ 120593) (119905) = (120601 ⊛ 120593) (119905) + (120595 ⊛ 120593) (119905)(iii) (120572120601 ⊛ 120595) (119905) = 120572 (120595 ⊛ 120601) (119905) 120572 is complex number(iv) ((120601 ⊛ 120595) ⊛ 120593) (119905) = (120601 ⊛ (120595 ⊛ 120593)) (119905)
By lowast we denote the convolution product defined by
(120601 lowast 120595) (119910) = intinfin
0
119911minus1120601 (119910119911minus1) 120595 (119911) 119889119911 (7)
By 120599(R+) denote the space of test functions of bounded
supports defined on R+that vanish more rapidly than every
power of 119909 as 119909 rarr infin Denote by 1205991the subset of 120599(R
+)
defined by
1205991= 120593 isin 120599 (R
+) intinfin
0
120593 (119909) 119889119909 = 1 (8)
Let Δ be the set of delta sequences satisfying the followingproperties
(1) 120575119899 isin 120599(R
+)
(2) intinfin0
120575119899(119909) 119889119909 = 1 119899 isin N
(3) intinfin0
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 lt infin 119899 isin N
(4) supp 120575119899(119909) sube [119886 119887
119899] 0 lt 119886
119899lt 119887119899and 119886119899 119887119899
rarr 0 as119899 rarr infin where
supp 120575119899(119909) = 119909 isin R
+ 120575119899(119909) = 0 forall119899 isin N (9)
Now we establish the following theorem
Theorem 1 Let 120601 120595 isin 120599(R+) then one has (120601 ⊛ 120595)
hh(119910) =
(120601hh lowast 120595) (119910)
Proof Let 120601 and 120595 be in 120599(R+) then we have
(120601 ⊛ 120595)hh
(119910) = intinfin
0
((120601 ⊛ 120595)o
h(119909) cos (119909119910)
+ (120601 ⊛ 120595)e
h(119909) sin (119909119910)) 119889119909
ie = intinfin
0
(intinfin
0
(120601 ⊛ 120595) (119905) sin (119909119905) 119889119905 cos (119909119910)
+ intinfin
0
(120601 ⊛ 120595) (119905) cos (119909119905) 119889119905
times sin (119909119910)) 119889119909
ie = intinfin
0
(intinfin
0
(intinfin
0
(120601 (119905119911minus1) sin (119909119905) cos (119909119910)
+ 120601 (119905119911minus1) cos (119909119905)
times sin (119909119910)) 119889119905) 119889119909)
times 120595 (119911) 119911minus1119889119911
(10)
By the change of variables 119905119911minus1 = 119908 we get that
(120601 ⊛ 120595)hh
(119910) = intinfin
0
(intinfin
0
(intinfin
0
(120601 (119908) sin (119909119911119908) cos (119909119910)
+ 120601 (119908) cos (119909119911119908)
times sin (119909119910)) 119889119908)119889119909)
times 120595 (119911) 119911 119889119911
ie = intinfin
0
(intinfin
0
(120601oh (119909119911) cos (119909119910)
+120601eh (119909119911) sin (119909119910)) 119889119909)
times 120595 (119911) 119911 119889119911
(11)
Once again by the change of variables 119909119911 = 119903 we get
(120601 ⊛ 120595)hh
(119910) = intinfin
0
119911minus1120601hh (119910119911minus1) 120595 (119911) 119889119911 (12)
This completes the proof of the theorem
Now we establish the following theorem
Theorem 2 Let 120601 120595 and 120579 be test functions in 120599(R+) then
120601 lowast (120595 ⊛ 120579) = (120601 lowast 120595) lowast 120579
Journal of Mathematics 3
Proof Let 119910 gt 0 be given Then from (6) and (7) we have
(120601 lowast (120595 ⊛ 120579)) (119910) = intinfin
0
119911minus1120601 (119910119911minus1) (120595 ⊛ 120579) (119911) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (intinfin
0
119905minus1120595 (119911119905minus1) 120579 (119905) 119889119905) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (120595 lowast 120579) (119911) 119889119911
(13)
Hence the theorem is completely proved
Theorem 3 Let 120593 120595 isin 1205991 then 120593 ⊛ 120595 = 120595 ⊛ 120593
Proof Proof of this theorem follows from technique similarto that of Theorem 2 Therefore we omit the details
Theorem 4 Let 1206011 1206012
isin 120599(R+) and 120593
1 1205932
isin 1205991 then the
following hold
(1) (1206011+ 1206012) lowast 1205931= 1206011lowast 1205931+ 1206012lowast 1205931
(2) if 120601119899
rarr 120601 isin 120599(R+) as 119899 rarr infin then 120601
119899lowast120593 rarr 120601lowast120593
as 119899 rarr infin
(3) 1206011lowast (1205931⊛ 1205932) = (120601
1lowast 1205931) lowast 1205932
(4) 1206011lowast 1205931isin 120599(R
+)
Proof Proof of (1) and (2) follows from elementary integralcalculus Proof of (3) is already obtained inTheorem 2 Proofof (4) is obvious by the properties of the product (7) Furtherdetails are thus avoided
This completes the proof of the theorem
Theorem 5 Let 120601 isin 120599(R+) and 120575
119899 isin Δ then 120601 lowast 120575
119899rarr 120601
as 119899 rarr infin
Proof Let 119870 be a compact set containing the support of 120601then by Property 2 of delta sequences we have
10038161003816100381610038161003816D119896
119910(120601 lowast 120575
119899minus 120601) (119910)
10038161003816100381610038161003816
le int119870
10038161003816100381610038161003816D119896
119910(119911minus1120601 (119910119911minus1) minus 120601 (119910))
100381610038161003816100381610038161003816100381610038161003816120575119899 (119910)
1003816100381610038161003816 119889119911(14)
Hence (14) goes to zero as 119899 rarr infinHence the theorem is completely proved
The Boehmian spaceB (120599 (1205991 ⊛) lowast Δ) is constructed
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899lowast 120595119899+ 119892119899lowast 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(15)
where 120572 isin C the field of complex numbers
The operation lowast and the differentiation are defined by
[119891119899
120596119899] lowast [
119892119899
120595119899] = [
119891119899lowast 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(16)
Similarly the spaceB (120599 (1205991 ⊛) ⊛ Δ) can be proved
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899⊛ 120595119899+ 119892119899⊛ 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(17)
The operation lowast and the differentiation are defined by
[119891119899
120596119899] ⊛ [
119892119899
120595119899] = [
119891119899⊛ 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(18)
As next let 120601 isin 120599(R+) then (120601)
hh(119910) isin 120599(R
+)
For some details we by definitions have that10038161003816100381610038161003816D119896
119910(120601)
hh(119910)
10038161003816100381610038161003816
le1
120587intinfin
0
D119896
119910(10038161003816100381610038161003816((120601)
o
h(119909) cos (119909119910) + (120601)
e
h(119909) sin (119909119910))
10038161003816100381610038161003816) 119889119909
le1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 +1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909
(19)
But since1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 =1
120587intinfin
0
int119887
119886
10038161003816100381610038161003816119909119896100381610038161003816100381610038161003816100381610038161003816120601 (119910)
1003816100381610038161003816 119889119910 119889119909
997888rarr 0
(20)
for every power of 119909 as 119909 rarr infin where [119886 119887] is a compact setcontaining the support of 120601 similarly we see that
1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909 997888rarr 0 (21)
for every power of 119909 as 119909 rarr infin Therefore we get that(120601)
hh(119910) isin 120599(R
+)
3 Hartley-Hilbert Transform of Quotients
Let [120601119899 120575119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) then we define the
extended Hartley-Hilbert transform of [120601119899 120575119899] as
([120601119899
120575119899])
hh
= [
[
(120601119899)hh
120575119899
]
]
(22)
in the quotient spaceB (120599 (1205991 ⊛) lowast Δ)
4 Journal of Mathematics
Theorem 6 The operator ([120601119899 120575119899])
hhis well defined and
linear fromB (120599 (1205991 ⊛) ⊛ Δ) intoB (120599 (120599
1 ⊛) lowast Δ)
Proof We show that ([120601119899 120575119899])
hhis well defined
Let [120595119899 120583119899] = [120585
119899 120598119899] in B (120599 (120599
1 ⊛) ⊛ Δ)
then by the concept of quotients of the spaceB (120599 (120599
1 ⊛) ⊛ Δ) we have
120595119899⊛ 120598119898
= 120585119898
⊛ 120583119899 (23)
Hence by Theorem 1 we get from (23) that
(120595119899)hhlowast 120598119898
= (120585119898)hhlowast 120583119899 (24)
Hence from (24) it follows that
(120595119899)hh
120583119899
sim(120585119899)hh
120598119899
(25)
inB (120599 (1205991 ⊛) lowast Δ)
Therefore
[
[
(120595119899)hh
120583119899
]
]
= [
[
(120585119899)hh
120598119899
]
]
(26)
inB (120599 (1205991 ⊛) lowast Δ)
That is
([120595119899
120583119899])
hh
= ([120585119899
120598119899])
hh
(27)
To show that ([120601119899 120575119899])
hhis linear let [119891
119899 120598119899]
[119892119899 120591119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) 120581 120578 isin R then
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= ([120581 119891119899
120598119899
] + [120578 119892119899
120591119899
])hh
ie = ([(120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899
120598119899⊛ 120591119899
])hh
ie = [
[
((120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899)hh
120598119899⊛ 120591119899
]
]
(28)
Linearity of the classical Hartley-Hilbert transformimplies
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899⊛ 120591119899)hh
+ 120578 (119892119899⊛ 120598119899)hh
120598119899 ⊛ 120591119899
]
]
(29)
Theorem 1 gives
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
lowast 120591119899 + 120578 (119892
119899)hh
lowast 120598119899
120598119899 ⊛ 120591119899
]
]
(30)
Thus by addition of Boehmians
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
120598119899
]
]
+ [
[
120578 (119892119899)hh
120591119899
]
]
(31)
Hence
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= 120581[
[
(119891119899)hh
120598119899
]
]
+ 120578[
[
(119892119899)hh
120591119899
]
]
(32)
The proof of the theorem is completed
Theorem 7 The necessary and sufficient condition for[119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) to be in the range of
([119892119899 120595119899])
hhis that 119892
119899belongs to range of the classical hh
for every 119899 isin N
Proof Let [119892119899 120595119899] be in the range of ([119892
119899 120595119899])
hh
then of course 119892119899belongs to the range of the classical hh forall119899 isin
NTo establish the converse let119892
119899be in the range of hhforall119899 isin
N Then there is 120601119899isin 120599(R
+) such that hh120601
119899= 119892119899 119899 isin N
Since [119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) we get 119892
119899lowast
120595119898
= 119892119898lowast 120595119899 forall119898 119899 isin N
Therefore Theorem 1 yields
hh (120601119899⊛ 120595119898) = hh (120601
119898⊛ 120595119899) forall119898 119899 isin N (33)
where 120601119899isin 120599(R
+) and 120595
119899 isin Δ forall119899 isin N
Thus 120601119899⊛ 120595119898
= 120601119898
⊛ 120595119899 119898 119899 isin N
Hence
[120601119899
120595119899] isin B (120599 (120599
1 ⊛) ⊛ Δ)
([120601119899
120595119899])
hh
= [119892119899
120595119899]
(34)
The theorem is therefore completely proved
Theorem 8 The transform ([120601119899 120595119899])
hhis consistent with
hh 120599(R+) rarr 120599(R
+)
Journal of Mathematics 5
Proof For every 120601 isin 120599(R+) let 120573 isin B (120599 (120599
1 ⊛) ⊛ Δ) be its
representative then we have 120573 = [120601 ⊛ 120595119899 120595119899] forall119899 isin N
120595119899 isin Δ For all 119899 isin N it is clear that 120595
119899 is independent
from the representativeByTheorem 1 we have
(120573)ℎℎ
= ([120601 ⊛ 120595
119899
120595119899
])hh
= [hh (120601 ⊛ 120595
119899)
120595119899
] = [hh120601 lowast 120595
119899
120595119899
]
(35)
which is the representative of hh120601 in the space 120599(R+)
Hence the proof of this theorem is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author expresses his gratitude to the referees for valuablesuggestions
References
[1] S K Q Al-Omari and A Kılıcman ldquoSome remarks on theextended Hartley-Hilbert and Fourier-Hilbert transforms ofboehmiansrdquo Abstract and Applied Analysis vol 2013 Article ID348701 6 pages 2013
[2] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[3] R P Millane ldquoAnalytical properties of the Hartley transformand their implicationsrdquo Proceedings of the IEEE vol 82 no 3pp 413ndash428 1994
[4] S K Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[5] S K Al-Omari ldquoHartley transforms on a certain space ofgeneralized functionsrdquo Georgian Mathematical Journal vol 20no 3 pp 415ndash426 2013
[6] S K Q Al-Omari and A Kilicman ldquoOn the generalizedHartley-Hilbert and Fourier-Hilbert transformsrdquo Advances inDifference Equations vol 2012 article 232 2012
[7] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[8] R Roopkumar ldquoMellin transform for Boehmiansrdquo Bulletin ofthe Institute of Mathematics Academia Sinica New Series vol4 no 1 pp 75ndash96 2009
[9] R Roopkumar ldquoGeneralized Radon transformrdquo Rocky Moun-tain Journal of Mathematics vol 36 no 4 pp 1375ndash1390 2006
[10] P Mikusinski and A Zayed ldquoThe Radon transform of Boehmi-ansrdquo Proceedings of the American Mathematical Society vol 118no 2 pp 561ndash570 1993
[11] S K Al-Omari and A Kilicman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2021 Article ID 405368 14 pages2012
[12] V Karunakaran and R Vembu ldquoHilbert transform on periodicboehmiansrdquo Houston Journal of Mathematics vol 29 no 2 pp437ndash454 2003
[13] V Karunakaran and R Roopkumar ldquoOperational calculus andFourier transform on Boehmiansrdquo Colloquium Mathematicumvol 102 no 1 pp 21ndash32 2005
[14] S K Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 6 pp 453ndash4622008
[15] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[16] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science AmsterdamThe Netherlands 1997
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
![Page 3: Research Article Some Definition of Hartley-Hilbert and ...downloads.hindawi.com/journals/jmath/2014/790161.pdf · andtostrongBoehmiansin[]. e Hartley-Hilbert and Fourier-Hilbert](https://reader035.fdocuments.in/reader035/viewer/2022062602/5edc9aa7ad6a402d666756a2/html5/thumbnails/3.jpg)
Journal of Mathematics 3
Proof Let 119910 gt 0 be given Then from (6) and (7) we have
(120601 lowast (120595 ⊛ 120579)) (119910) = intinfin
0
119911minus1120601 (119910119911minus1) (120595 ⊛ 120579) (119911) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (intinfin
0
119905minus1120595 (119911119905minus1) 120579 (119905) 119889119905) 119889119911
ie = intinfin
0
119911minus1120601 (119910119911minus1) (120595 lowast 120579) (119911) 119889119911
(13)
Hence the theorem is completely proved
Theorem 3 Let 120593 120595 isin 1205991 then 120593 ⊛ 120595 = 120595 ⊛ 120593
Proof Proof of this theorem follows from technique similarto that of Theorem 2 Therefore we omit the details
Theorem 4 Let 1206011 1206012
isin 120599(R+) and 120593
1 1205932
isin 1205991 then the
following hold
(1) (1206011+ 1206012) lowast 1205931= 1206011lowast 1205931+ 1206012lowast 1205931
(2) if 120601119899
rarr 120601 isin 120599(R+) as 119899 rarr infin then 120601
119899lowast120593 rarr 120601lowast120593
as 119899 rarr infin
(3) 1206011lowast (1205931⊛ 1205932) = (120601
1lowast 1205931) lowast 1205932
(4) 1206011lowast 1205931isin 120599(R
+)
Proof Proof of (1) and (2) follows from elementary integralcalculus Proof of (3) is already obtained inTheorem 2 Proofof (4) is obvious by the properties of the product (7) Furtherdetails are thus avoided
This completes the proof of the theorem
Theorem 5 Let 120601 isin 120599(R+) and 120575
119899 isin Δ then 120601 lowast 120575
119899rarr 120601
as 119899 rarr infin
Proof Let 119870 be a compact set containing the support of 120601then by Property 2 of delta sequences we have
10038161003816100381610038161003816D119896
119910(120601 lowast 120575
119899minus 120601) (119910)
10038161003816100381610038161003816
le int119870
10038161003816100381610038161003816D119896
119910(119911minus1120601 (119910119911minus1) minus 120601 (119910))
100381610038161003816100381610038161003816100381610038161003816120575119899 (119910)
1003816100381610038161003816 119889119911(14)
Hence (14) goes to zero as 119899 rarr infinHence the theorem is completely proved
The Boehmian spaceB (120599 (1205991 ⊛) lowast Δ) is constructed
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899lowast 120595119899+ 119892119899lowast 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(15)
where 120572 isin C the field of complex numbers
The operation lowast and the differentiation are defined by
[119891119899
120596119899] lowast [
119892119899
120595119899] = [
119891119899lowast 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(16)
Similarly the spaceB (120599 (1205991 ⊛) ⊛ Δ) can be proved
The sum of two Boehmians andmultiplication by a scalarcan be defined in a natural way
[119891119899
120596119899] + [
119892119899
120595119899] = [
119891119899⊛ 120595119899+ 119892119899⊛ 120596119899
120596119899⊛ 120595119899
]
120572 [119891119899
120596119899] = [120572
119891119899
120596119899] = [
120572119891119899
120596119899]
(17)
The operation lowast and the differentiation are defined by
[119891119899
120596119899] ⊛ [
119892119899
120595119899] = [
119891119899⊛ 119892119899
120596119899⊛ 120595119899]
D120572 [
119891119899
120596119899] = [
D120572119891119899
120596119899
]
(18)
As next let 120601 isin 120599(R+) then (120601)
hh(119910) isin 120599(R
+)
For some details we by definitions have that10038161003816100381610038161003816D119896
119910(120601)
hh(119910)
10038161003816100381610038161003816
le1
120587intinfin
0
D119896
119910(10038161003816100381610038161003816((120601)
o
h(119909) cos (119909119910) + (120601)
e
h(119909) sin (119909119910))
10038161003816100381610038161003816) 119889119909
le1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 +1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909
(19)
But since1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
o
h(119909)
10038161003816100381610038161003816 119889119909 =1
120587intinfin
0
int119887
119886
10038161003816100381610038161003816119909119896100381610038161003816100381610038161003816100381610038161003816120601 (119910)
1003816100381610038161003816 119889119910 119889119909
997888rarr 0
(20)
for every power of 119909 as 119909 rarr infin where [119886 119887] is a compact setcontaining the support of 120601 similarly we see that
1
120587intinfin
0
100381610038161003816100381610038161199091198961003816100381610038161003816100381610038161003816100381610038161003816(120601)
e
h(119909)
10038161003816100381610038161003816 119889119909 997888rarr 0 (21)
for every power of 119909 as 119909 rarr infin Therefore we get that(120601)
hh(119910) isin 120599(R
+)
3 Hartley-Hilbert Transform of Quotients
Let [120601119899 120575119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) then we define the
extended Hartley-Hilbert transform of [120601119899 120575119899] as
([120601119899
120575119899])
hh
= [
[
(120601119899)hh
120575119899
]
]
(22)
in the quotient spaceB (120599 (1205991 ⊛) lowast Δ)
4 Journal of Mathematics
Theorem 6 The operator ([120601119899 120575119899])
hhis well defined and
linear fromB (120599 (1205991 ⊛) ⊛ Δ) intoB (120599 (120599
1 ⊛) lowast Δ)
Proof We show that ([120601119899 120575119899])
hhis well defined
Let [120595119899 120583119899] = [120585
119899 120598119899] in B (120599 (120599
1 ⊛) ⊛ Δ)
then by the concept of quotients of the spaceB (120599 (120599
1 ⊛) ⊛ Δ) we have
120595119899⊛ 120598119898
= 120585119898
⊛ 120583119899 (23)
Hence by Theorem 1 we get from (23) that
(120595119899)hhlowast 120598119898
= (120585119898)hhlowast 120583119899 (24)
Hence from (24) it follows that
(120595119899)hh
120583119899
sim(120585119899)hh
120598119899
(25)
inB (120599 (1205991 ⊛) lowast Δ)
Therefore
[
[
(120595119899)hh
120583119899
]
]
= [
[
(120585119899)hh
120598119899
]
]
(26)
inB (120599 (1205991 ⊛) lowast Δ)
That is
([120595119899
120583119899])
hh
= ([120585119899
120598119899])
hh
(27)
To show that ([120601119899 120575119899])
hhis linear let [119891
119899 120598119899]
[119892119899 120591119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) 120581 120578 isin R then
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= ([120581 119891119899
120598119899
] + [120578 119892119899
120591119899
])hh
ie = ([(120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899
120598119899⊛ 120591119899
])hh
ie = [
[
((120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899)hh
120598119899⊛ 120591119899
]
]
(28)
Linearity of the classical Hartley-Hilbert transformimplies
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899⊛ 120591119899)hh
+ 120578 (119892119899⊛ 120598119899)hh
120598119899 ⊛ 120591119899
]
]
(29)
Theorem 1 gives
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
lowast 120591119899 + 120578 (119892
119899)hh
lowast 120598119899
120598119899 ⊛ 120591119899
]
]
(30)
Thus by addition of Boehmians
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
120598119899
]
]
+ [
[
120578 (119892119899)hh
120591119899
]
]
(31)
Hence
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= 120581[
[
(119891119899)hh
120598119899
]
]
+ 120578[
[
(119892119899)hh
120591119899
]
]
(32)
The proof of the theorem is completed
Theorem 7 The necessary and sufficient condition for[119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) to be in the range of
([119892119899 120595119899])
hhis that 119892
119899belongs to range of the classical hh
for every 119899 isin N
Proof Let [119892119899 120595119899] be in the range of ([119892
119899 120595119899])
hh
then of course 119892119899belongs to the range of the classical hh forall119899 isin
NTo establish the converse let119892
119899be in the range of hhforall119899 isin
N Then there is 120601119899isin 120599(R
+) such that hh120601
119899= 119892119899 119899 isin N
Since [119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) we get 119892
119899lowast
120595119898
= 119892119898lowast 120595119899 forall119898 119899 isin N
Therefore Theorem 1 yields
hh (120601119899⊛ 120595119898) = hh (120601
119898⊛ 120595119899) forall119898 119899 isin N (33)
where 120601119899isin 120599(R
+) and 120595
119899 isin Δ forall119899 isin N
Thus 120601119899⊛ 120595119898
= 120601119898
⊛ 120595119899 119898 119899 isin N
Hence
[120601119899
120595119899] isin B (120599 (120599
1 ⊛) ⊛ Δ)
([120601119899
120595119899])
hh
= [119892119899
120595119899]
(34)
The theorem is therefore completely proved
Theorem 8 The transform ([120601119899 120595119899])
hhis consistent with
hh 120599(R+) rarr 120599(R
+)
Journal of Mathematics 5
Proof For every 120601 isin 120599(R+) let 120573 isin B (120599 (120599
1 ⊛) ⊛ Δ) be its
representative then we have 120573 = [120601 ⊛ 120595119899 120595119899] forall119899 isin N
120595119899 isin Δ For all 119899 isin N it is clear that 120595
119899 is independent
from the representativeByTheorem 1 we have
(120573)ℎℎ
= ([120601 ⊛ 120595
119899
120595119899
])hh
= [hh (120601 ⊛ 120595
119899)
120595119899
] = [hh120601 lowast 120595
119899
120595119899
]
(35)
which is the representative of hh120601 in the space 120599(R+)
Hence the proof of this theorem is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author expresses his gratitude to the referees for valuablesuggestions
References
[1] S K Q Al-Omari and A Kılıcman ldquoSome remarks on theextended Hartley-Hilbert and Fourier-Hilbert transforms ofboehmiansrdquo Abstract and Applied Analysis vol 2013 Article ID348701 6 pages 2013
[2] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[3] R P Millane ldquoAnalytical properties of the Hartley transformand their implicationsrdquo Proceedings of the IEEE vol 82 no 3pp 413ndash428 1994
[4] S K Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[5] S K Al-Omari ldquoHartley transforms on a certain space ofgeneralized functionsrdquo Georgian Mathematical Journal vol 20no 3 pp 415ndash426 2013
[6] S K Q Al-Omari and A Kilicman ldquoOn the generalizedHartley-Hilbert and Fourier-Hilbert transformsrdquo Advances inDifference Equations vol 2012 article 232 2012
[7] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[8] R Roopkumar ldquoMellin transform for Boehmiansrdquo Bulletin ofthe Institute of Mathematics Academia Sinica New Series vol4 no 1 pp 75ndash96 2009
[9] R Roopkumar ldquoGeneralized Radon transformrdquo Rocky Moun-tain Journal of Mathematics vol 36 no 4 pp 1375ndash1390 2006
[10] P Mikusinski and A Zayed ldquoThe Radon transform of Boehmi-ansrdquo Proceedings of the American Mathematical Society vol 118no 2 pp 561ndash570 1993
[11] S K Al-Omari and A Kilicman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2021 Article ID 405368 14 pages2012
[12] V Karunakaran and R Vembu ldquoHilbert transform on periodicboehmiansrdquo Houston Journal of Mathematics vol 29 no 2 pp437ndash454 2003
[13] V Karunakaran and R Roopkumar ldquoOperational calculus andFourier transform on Boehmiansrdquo Colloquium Mathematicumvol 102 no 1 pp 21ndash32 2005
[14] S K Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 6 pp 453ndash4622008
[15] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[16] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science AmsterdamThe Netherlands 1997
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
![Page 4: Research Article Some Definition of Hartley-Hilbert and ...downloads.hindawi.com/journals/jmath/2014/790161.pdf · andtostrongBoehmiansin[]. e Hartley-Hilbert and Fourier-Hilbert](https://reader035.fdocuments.in/reader035/viewer/2022062602/5edc9aa7ad6a402d666756a2/html5/thumbnails/4.jpg)
4 Journal of Mathematics
Theorem 6 The operator ([120601119899 120575119899])
hhis well defined and
linear fromB (120599 (1205991 ⊛) ⊛ Δ) intoB (120599 (120599
1 ⊛) lowast Δ)
Proof We show that ([120601119899 120575119899])
hhis well defined
Let [120595119899 120583119899] = [120585
119899 120598119899] in B (120599 (120599
1 ⊛) ⊛ Δ)
then by the concept of quotients of the spaceB (120599 (120599
1 ⊛) ⊛ Δ) we have
120595119899⊛ 120598119898
= 120585119898
⊛ 120583119899 (23)
Hence by Theorem 1 we get from (23) that
(120595119899)hhlowast 120598119898
= (120585119898)hhlowast 120583119899 (24)
Hence from (24) it follows that
(120595119899)hh
120583119899
sim(120585119899)hh
120598119899
(25)
inB (120599 (1205991 ⊛) lowast Δ)
Therefore
[
[
(120595119899)hh
120583119899
]
]
= [
[
(120585119899)hh
120598119899
]
]
(26)
inB (120599 (1205991 ⊛) lowast Δ)
That is
([120595119899
120583119899])
hh
= ([120585119899
120598119899])
hh
(27)
To show that ([120601119899 120575119899])
hhis linear let [119891
119899 120598119899]
[119892119899 120591119899] isin B (120599 (120599
1 ⊛) ⊛ Δ) 120581 120578 isin R then
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= ([120581 119891119899
120598119899
] + [120578 119892119899
120591119899
])hh
ie = ([(120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899
120598119899⊛ 120591119899
])hh
ie = [
[
((120581119891119899) ⊛ 120591119899+ (120578119892119899) ⊛ 120598119899)hh
120598119899⊛ 120591119899
]
]
(28)
Linearity of the classical Hartley-Hilbert transformimplies
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899⊛ 120591119899)hh
+ 120578 (119892119899⊛ 120598119899)hh
120598119899 ⊛ 120591119899
]
]
(29)
Theorem 1 gives
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
lowast 120591119899 + 120578 (119892
119899)hh
lowast 120598119899
120598119899 ⊛ 120591119899
]
]
(30)
Thus by addition of Boehmians
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= [
[
120581 (119891119899)hh
120598119899
]
]
+ [
[
120578 (119892119899)hh
120591119899
]
]
(31)
Hence
(120581[119891119899
120598119899] + 120578 [
119892119899
120591119899])
hh
= 120581[
[
(119891119899)hh
120598119899
]
]
+ 120578[
[
(119892119899)hh
120591119899
]
]
(32)
The proof of the theorem is completed
Theorem 7 The necessary and sufficient condition for[119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) to be in the range of
([119892119899 120595119899])
hhis that 119892
119899belongs to range of the classical hh
for every 119899 isin N
Proof Let [119892119899 120595119899] be in the range of ([119892
119899 120595119899])
hh
then of course 119892119899belongs to the range of the classical hh forall119899 isin
NTo establish the converse let119892
119899be in the range of hhforall119899 isin
N Then there is 120601119899isin 120599(R
+) such that hh120601
119899= 119892119899 119899 isin N
Since [119892119899 120595119899] isin B (120599 (120599
1 ⊛) lowast Δ) we get 119892
119899lowast
120595119898
= 119892119898lowast 120595119899 forall119898 119899 isin N
Therefore Theorem 1 yields
hh (120601119899⊛ 120595119898) = hh (120601
119898⊛ 120595119899) forall119898 119899 isin N (33)
where 120601119899isin 120599(R
+) and 120595
119899 isin Δ forall119899 isin N
Thus 120601119899⊛ 120595119898
= 120601119898
⊛ 120595119899 119898 119899 isin N
Hence
[120601119899
120595119899] isin B (120599 (120599
1 ⊛) ⊛ Δ)
([120601119899
120595119899])
hh
= [119892119899
120595119899]
(34)
The theorem is therefore completely proved
Theorem 8 The transform ([120601119899 120595119899])
hhis consistent with
hh 120599(R+) rarr 120599(R
+)
Journal of Mathematics 5
Proof For every 120601 isin 120599(R+) let 120573 isin B (120599 (120599
1 ⊛) ⊛ Δ) be its
representative then we have 120573 = [120601 ⊛ 120595119899 120595119899] forall119899 isin N
120595119899 isin Δ For all 119899 isin N it is clear that 120595
119899 is independent
from the representativeByTheorem 1 we have
(120573)ℎℎ
= ([120601 ⊛ 120595
119899
120595119899
])hh
= [hh (120601 ⊛ 120595
119899)
120595119899
] = [hh120601 lowast 120595
119899
120595119899
]
(35)
which is the representative of hh120601 in the space 120599(R+)
Hence the proof of this theorem is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author expresses his gratitude to the referees for valuablesuggestions
References
[1] S K Q Al-Omari and A Kılıcman ldquoSome remarks on theextended Hartley-Hilbert and Fourier-Hilbert transforms ofboehmiansrdquo Abstract and Applied Analysis vol 2013 Article ID348701 6 pages 2013
[2] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[3] R P Millane ldquoAnalytical properties of the Hartley transformand their implicationsrdquo Proceedings of the IEEE vol 82 no 3pp 413ndash428 1994
[4] S K Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[5] S K Al-Omari ldquoHartley transforms on a certain space ofgeneralized functionsrdquo Georgian Mathematical Journal vol 20no 3 pp 415ndash426 2013
[6] S K Q Al-Omari and A Kilicman ldquoOn the generalizedHartley-Hilbert and Fourier-Hilbert transformsrdquo Advances inDifference Equations vol 2012 article 232 2012
[7] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[8] R Roopkumar ldquoMellin transform for Boehmiansrdquo Bulletin ofthe Institute of Mathematics Academia Sinica New Series vol4 no 1 pp 75ndash96 2009
[9] R Roopkumar ldquoGeneralized Radon transformrdquo Rocky Moun-tain Journal of Mathematics vol 36 no 4 pp 1375ndash1390 2006
[10] P Mikusinski and A Zayed ldquoThe Radon transform of Boehmi-ansrdquo Proceedings of the American Mathematical Society vol 118no 2 pp 561ndash570 1993
[11] S K Al-Omari and A Kilicman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2021 Article ID 405368 14 pages2012
[12] V Karunakaran and R Vembu ldquoHilbert transform on periodicboehmiansrdquo Houston Journal of Mathematics vol 29 no 2 pp437ndash454 2003
[13] V Karunakaran and R Roopkumar ldquoOperational calculus andFourier transform on Boehmiansrdquo Colloquium Mathematicumvol 102 no 1 pp 21ndash32 2005
[14] S K Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 6 pp 453ndash4622008
[15] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[16] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science AmsterdamThe Netherlands 1997
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
![Page 5: Research Article Some Definition of Hartley-Hilbert and ...downloads.hindawi.com/journals/jmath/2014/790161.pdf · andtostrongBoehmiansin[]. e Hartley-Hilbert and Fourier-Hilbert](https://reader035.fdocuments.in/reader035/viewer/2022062602/5edc9aa7ad6a402d666756a2/html5/thumbnails/5.jpg)
Journal of Mathematics 5
Proof For every 120601 isin 120599(R+) let 120573 isin B (120599 (120599
1 ⊛) ⊛ Δ) be its
representative then we have 120573 = [120601 ⊛ 120595119899 120595119899] forall119899 isin N
120595119899 isin Δ For all 119899 isin N it is clear that 120595
119899 is independent
from the representativeByTheorem 1 we have
(120573)ℎℎ
= ([120601 ⊛ 120595
119899
120595119899
])hh
= [hh (120601 ⊛ 120595
119899)
120595119899
] = [hh120601 lowast 120595
119899
120595119899
]
(35)
which is the representative of hh120601 in the space 120599(R+)
Hence the proof of this theorem is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author expresses his gratitude to the referees for valuablesuggestions
References
[1] S K Q Al-Omari and A Kılıcman ldquoSome remarks on theextended Hartley-Hilbert and Fourier-Hilbert transforms ofboehmiansrdquo Abstract and Applied Analysis vol 2013 Article ID348701 6 pages 2013
[2] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[3] R P Millane ldquoAnalytical properties of the Hartley transformand their implicationsrdquo Proceedings of the IEEE vol 82 no 3pp 413ndash428 1994
[4] S K Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[5] S K Al-Omari ldquoHartley transforms on a certain space ofgeneralized functionsrdquo Georgian Mathematical Journal vol 20no 3 pp 415ndash426 2013
[6] S K Q Al-Omari and A Kilicman ldquoOn the generalizedHartley-Hilbert and Fourier-Hilbert transformsrdquo Advances inDifference Equations vol 2012 article 232 2012
[7] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[8] R Roopkumar ldquoMellin transform for Boehmiansrdquo Bulletin ofthe Institute of Mathematics Academia Sinica New Series vol4 no 1 pp 75ndash96 2009
[9] R Roopkumar ldquoGeneralized Radon transformrdquo Rocky Moun-tain Journal of Mathematics vol 36 no 4 pp 1375ndash1390 2006
[10] P Mikusinski and A Zayed ldquoThe Radon transform of Boehmi-ansrdquo Proceedings of the American Mathematical Society vol 118no 2 pp 561ndash570 1993
[11] S K Al-Omari and A Kilicman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2021 Article ID 405368 14 pages2012
[12] V Karunakaran and R Vembu ldquoHilbert transform on periodicboehmiansrdquo Houston Journal of Mathematics vol 29 no 2 pp437ndash454 2003
[13] V Karunakaran and R Roopkumar ldquoOperational calculus andFourier transform on Boehmiansrdquo Colloquium Mathematicumvol 102 no 1 pp 21ndash32 2005
[14] S K Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 6 pp 453ndash4622008
[15] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[16] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science AmsterdamThe Netherlands 1997
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
![Page 6: Research Article Some Definition of Hartley-Hilbert and ...downloads.hindawi.com/journals/jmath/2014/790161.pdf · andtostrongBoehmiansin[]. e Hartley-Hilbert and Fourier-Hilbert](https://reader035.fdocuments.in/reader035/viewer/2022062602/5edc9aa7ad6a402d666756a2/html5/thumbnails/6.jpg)
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
