Approximationbymodiﬁed Kantorovich–Szásztypeoperatorsinvolving …... · 2020. 5. 1. ·...

Ansari et al. Advances in Difference Equations (2020) 2020:192 https://doi.org/10.1186/s13662-020-02645-6

R E S E A R C H Open Access

Approximation by modifiedKantorovich–Szász type operators involvingCharlier polynomialsK.J. Ansari1, M. Mursaleen2,3,4*, M. Shareef KP5 and M. Ghouse6

*Correspondence:[email protected] of Medical Research,China Medical University Hospital,China Medical University (Taiwan),Taichung, Taiwan3Department of Mathematics,Aligarh Muslim University, Aligarh,IndiaFull list of author information isavailable at the end of the article

AbstractIn this paper, we give some direct approximation results by modifiedKantorovich–Szász type operators involving Charlier polynomials. Further,approximation results are also developed in polynomial weighted spaces. Moreover,for the functions of bounded variation, approximation results are proved. Finally,some graphical examples are provided to show comparisons of convergencebetween old and modified operators towards a function under different parametersand conditions.

MSC: 41A10; 41A25; 41A36

Keywords: Szász operator; Charlier polynomials; Modulus of continuity; Rate ofconvergence

1 IntroductionIn 1950, Szász [20] introduced positive linear operators in the sense of exponential growthon nonnegative semiaxes and exhaustively investigated them. These operators later be-came known as Szász operators. The Szász type operators involving Charlier polynomialswere defined in [21] as

Ln(f ; y, b) = e–1(

1 –1b

)(b–1)ny ∞∑l=0

C(b)l (–(b – 1)ny)

l!f(

ln

), (1.1)

where b > 1 and y ≥ 0, having the generating functions [5] of the form

et(

1 –tb

)u

=∞∑l=0

C(b)l (u)

tl

l!, |t| < b, (1.2)

where C(b)l (u) =

∑lr=0( l

r )(–u)r( 1b )r and (k)0 = 1, (k)m = k(k + 1) · · · (k + m – 1), for m ≥ 1.

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Motivated by the work done in [9], we define the Kantorovich generalization [10] of (1.2)as follows:

Q(μn ,νn)n,b (f ; y) = νne–1

(1 –

1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!

∫ l+1/νn

l/νn

f (t) dt (1.3)

where μn and νn are sequences of positive numbers which are increasing and unboundedsuch that

limn→∞

1νn

= 0,μn

νn= 1 + O

(1νn

). (1.4)

If we take μn = νn = n, we will have the operators defined in [9].For some recent and interesting results on the various generalizations and correspond-

ing approximation results, we refer to [1, 3, 6–8, 14–17, 22].

2 Auxiliary resultsWe first present some auxiliary results.

Lemma 2.1 Let Q(μn ,νn)n,b be defined by (1.3). Then, we have

1. Q(μn ,νn)n,b (1; y) = 1,

2. Q(μn ,νn)n,b (t; y) = μn

νny + 3

2νn

3. Q(μn ,νn)n,b (t2; y) = μ2

nν2

ny2 + μn

ν2n

(4 + 1b–1 )y + 10

3ν2n

.

4. Q(μn ,νn)n,b (t3; y) = μ3

nν3

ny3 + μ2

nν3

n( 15

2 + 3b–1 )y2 + μn

ν3n

( 312 + 15

2(b–1) + 2(b–1)2 )y + 37

4ν3n

.

5. Q(μn ,νn)n,b (t4; y) =

μ4n

ν4n

y4 + μ3n

ν4n

(12 + 6b–1 )y3 + μ2

nν4

n(46 + 36

b–1 + 11(b–1)2 )y2 + μn

ν4n

(64 + 46b–1 + 24

(b–1)2 + 6(b–1)3 )y + 151

5ν4n

.

Proof With the help of the Charlier polynomials’ generating function given by (1.2), aftersome simple calculations, we obtain

∞∑l=0

C(b)l (–(b – 1)μny)

l!= e

(1 –

1b

)–(b–1)μny

,

∞∑l=0

lC(b)l (–(b – 1)μny)

l!= e

(1 –

1b

)–(b–1)μny

(1 + μny),

∞∑l=0

l2C(b)l (–(b – 1)μny)

l!= e

(1 –

1b

)–(b–1)μny(μ2

ny2 + μny(

3 +1

b – 1

)+ 2

),

∞∑l=0

l3C(b)l (–(b – 1)μny)

l!= e

(1 –

1b

)–(b–1)μny(μ3

ny3 + μ2n

(6 +

3b – 1

)y2

+ μn

(10 +

6b – 1

+2

(b – 1)2

)y + 5

),


∞∑l=0

l4C(b)l (–(b – 1)μny)

l!= e

(1 –

1b

)–(b–1)μny(μ4

ny4 + μ3n

(10 +

6b – 1

)y3

+ μ2n

(32 +

30b – 1

+11

(b – 1)2

)y2

+ μn

(37 +

32b – 1

+20

(b – 1)2 +6

(b – 1)3

)y + 15

).

From the above equalities, the claims of the lemma can be obtained. �

Lemma 2.2 For the operator Q(μn ,νn)n,b given by (1.3), we have the following equalities:

(i) Q(μn ,νn)n,b (t – y; y) =

(μn

νn– 1

)y +

1νn

,

(ii) Q(μn ,νn)n,b

((t – y)2; y

)=

(μn

νn– 1

)2

y2 +(

μn

ν2n

(3 +

1b – 1

)–

2νn

)y +

2ν2

n,

(iii) Q(μn ,νn)n,b

((t – y)4; y

)

=(

μn

νn– 1

)4

y4

+ 2(

μ3n

ν4n

(5 +

3b – 1

)–

6μ2n

ν3n

(2 +

1b – 1

)+

3μn

ν2n

(3 +

1b – 1

)–

2νn

)y3

+(

μ2n

ν4n

(32 +

30b – 1

+11

(b – 1)2

)–

4μn

ν3n

(10 +

6b – 1

+2

(b – 1)2

)+

12ν2

n

)y2

+(

μn

ν4n

(37 +

32b – 1

+20

(b – 1)2 +6

(b – 1)3

)–

20ν3

n

)y +

15ν4

n.

3 Local approximation resultsIn what follows, let Q(μn ,νn)

n,b (t – y; y) = χμn ,νn (y) and Q(μn ,νn)n,b ((t – y)2; y) = ξμn ,νn (y). We will

now give two theorems on the uniform convergence and the order of approximation.

Theorem 3.1 Let f ∈ C[0,∞) ∩ G. Then limn→∞ Q(μn ,νn)n,b (f ; y) = f (y), the sequence of oper-

ators Eq. (1.3) converges uniformly in each compact subset of [0,∞), where

G :={

f : R+0 →R,

∣∣F(y)∣∣ =

∣∣∣∣∫ y

0f (s) ds

∣∣∣∣ ≤ KeBy, K ∈ R+ and B ∈R

}.

Proof From Lemma 2.1(1)–(3), we get

limn→∞Q(μn ,νn)

n,b(sk ; y

)= yk , k = 0, 1, 2.

The proof of the theorem is established by taking advantage of the above uniform conver-gence in each compact subset of [0,∞) and the famous Korovkin’s theorem. �

Suppose f ∈ C[0,∞), i.e., f belongs to the space of uniformly continuous functions on[0,∞). If δ > 0, then the modulus of continuity ω(f , δ) is defined by

ω(f , δ) := supσ ,ς∈[0,∞)|σ–ς |≤δ

∣∣f (σ ) – f (ς )∣∣.


Theorem 3.2 Let f ∈ C[0,∞) ∩ E. For the operators Q(μn ,νn)n,b (f ; y) given by (1.3) the follow-

ing estimate holds:

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣

≤{

1 +

√(μn – νn)2y2 +

((4 +

1b – 1

)μn – 3νn

)y +

103

}ω

(f ,

1νn

). (3.1)

Proof From (1.3) and the property of modulus of continuity, the left-hand side of (3.1)leads to

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣

≤ νne–1(

1 –1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!

∫ l+1/νn

l/νn

∣∣f (t) – f (y)∣∣dt

≤{

1 +1δνne–1

(1 –

1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!

∫ l+1/νn

l/νn

|t – y|dt

}ω(f , δ).

Using Cauchy–Schwarz inequality for the integral, we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣

≤{

1 +1δ

e–1(

1 –1b

)(b–1)μny

×∞∑l=0

C(b)l (–(b – 1)μny)

l!

(νn

∫ l+1/νn

l/νn

(t – y)2 dt)1/2

}ω(f , δ). (3.2)

In the above sum, we apply Cauchy–Schwarz inequality, and then in view of Lemma 2.1,(3.2) becomes

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣

≤{

1 +1δ

(e–1

(1 –

1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!

)1/2

×(

νne–1(

1 –1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!

∫ l+1/νn

l/νn

(t – y)2 dt

)1/2}ω(f , δ)

≤{

1 +1δ

(Q(μn ,νn)

n,b((t – y)2; y, b

))1/2}ω(f , δ)

≤{

1 +1δ

1νn

√(μn – νn)2y2 +

((4 +

1b – 1

)μn – 3νn

)y +

103

}ω

(f ,

1δ

),

where, taking δ = 1νn

, we get (3.1). �


Let a1, a2 > 0 be fixed. We now consider the following space of Lipschitz type (see[18]):

Lip(a1,a2)M (r) :=

{f ∈ C[0,∞) :

∣∣f (κ) – f (λ)∣∣ ≤ M

|κ – λ|r(κ + a1λ2 + a2λ)r/2 ;λ,κ > 0

}, (3.3)

where M is a positive constant and 0 < r ≤ 1.

Theorem 3.3 Let f ∈ Lip(a1,a2)M (r) and r ∈ (0, 1], then ∀y > 0, we have

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ M(

ξμn ,νn (y)(a1y2 + a2y)

)r/2

.

Proof Since

∣∣f (t) – f (y)∣∣ ≤ M

|t – y|r(t + a1y2 + a2y)r/2 ,

one has

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ MQ(μn ,νn)n,b

( |t – y|r(t + a1y2 + a2y)r/2 ; y

).

Applying Hölder’s inequality with p = 2r and 2

2–r , we find that

∣∣Q(μn ,νn)n,b (f ; y) – f (y)


((t – y)2

t + a1y2 + a2y; y

)r/2

.

Since f ∈ Lip(a1,a2)M (r) and 1

t+a1y2+a2y < 1a1y2+a2y , ∀y ∈ (0,∞), we have

∣∣Q(μn ,νn)n,b (f ; y) – f (y)


((t – y)2

a1y2 + a2y; y

)r/2

≤ M(a1y2 + a2y)r/2 Q

(μn ,νn)n,b

((t – y)2; y

)r/2

= M(

ξμn ,νn (y)(a1y2 + a2y)

)r/2

.

Our proof is now completed. �

We denote the space of all functions h on [0, 1) which are real-valued, uniformly contin-uous, as well as bounded by CB[0,∞) and endow it with the norm ‖h‖∞ = supy∈[0,1) |h(y)|.Further, we obtain a local direct estimate for the operators (1.3), using the Lipschitz max-imal function of order r introduced by Lenze [13] as:

ωr(h, y) = supt =y,t∈[0,∞)

|h(t) – h(y)||t – y|r , (3.4)

where y ∈ [0, 1) and r ∈ (0, 1].


Theorem 3.4 Let f ∈ CB[0,∞) and 0 < r ≤ 1, then ∀y ∈ [0,∞)

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ ωr(f , y)(ξμn ,νn (y)

)r .

Proof By equation (3.4),

∣∣f (t) – f (y)∣∣ ≤ ωr(f , y)|t – y|r .

Applying Q(μn ,νn)n,b on both sides of the above inequality, then using Lemma 2.1, as well as

Hölder’s inequality with p = 2/r, q = 2/(2 – r), we obtain

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ ωr(f , y)Q(μn ,νn)n,b

(|t – y|r ; y)

≤ ωr(f , y)(Q(μn ,νn)

n,b((t – y)2; y

))r/2

= ωr(f , y)(ξμn ,νn (y)

)r/2.

Thus, we have our desired result. �

The Peetre’s K-functional is given by

K(g, δ) = infh∈C2

B[0,∞)

{‖g – h‖∞ + δ‖h‖C2B

},

where C2B[0,∞) = {h ∈ CB[0,∞) : h′, h′′ ∈ CB[0,∞)} with the norm ‖h‖C2

B= ‖h‖∞ +‖h′‖∞ +

‖h′′‖∞. Also, the inequality

K(g, δ) ≤ M(ω2(g,

√δ) + min{1, δ}‖g‖∞

)

holds for all δ > 0, where ω2 is the second-order modulus of smoothness of g ∈ CB[0,∞),which is defined by

ω2(g, δ) = sup0<|h|≤δ

supy,y+2h∈[0,∞)

∣∣g(y + 2h) – 2g(y + h) + g(y)∣∣.

Theorem 3.5 If f ∈ CB[0,∞), then

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ 4K(f , ζμn ,νn (y)) + ω(f ,χμn ,νn (y)),

where ζμn ,νn (y) = (ξμn ,νn (y) + χ2μn ,νn (y))/4. Furthermore,

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ M(ω2

(f ,

√ζμn ,νn (y)

)+ min{1, ζμn ,νn (y)}‖f ‖∞

)+ω

(f ,

∣∣χμn ,νn (y)∣∣).

Proof For f ∈ CB[0,∞), we define the auxiliary operator as follows:

Q(μn ,νn)n,b (f ; y) = Q(μn ,νn)

n,b (f ; y) – f(

μn

νny +

32νn

)+ f (y). (3.5)


After taking the absolute value of both sides,

∣∣Q(μn ,νn)n,b (f ; y)

∣∣ ≤ ∣∣Q(μn ,νn)n,b (f ; y)

∣∣ +∣∣∣∣f

(μn

νny +

32νn

)∣∣∣∣ +∣∣f (y)

∣∣≤ ‖f ‖∞

∣∣Q(μn ,νn)n,b (1; y, b)

∣∣ + ‖f ‖∞ + ‖f ‖∞

≤ 3‖f ‖∞. (3.6)

By Lemma 2.1, we have Q(μn ,νn)n,b (t; y) = y, and therefore Q(μn ,νn)

n,b (t – y; y) = 0.Let g ∈ C2

B[0,∞), using Taylor’s theorem, we can write

g(t) = g(y) + g ′(y)(t – y) +∫ t

y(t – u)g ′′(u) du.

Applying operator Q(μn ,νn)n,b to the above equation, we get

Q(μn ,νn)n,b (g; y) – g(y)

= Q(μn ,νn)n,b

(∫ t

y(t – u)g ′′(u) du; y

)

= Q(μn ,νn)n,b

(∫ t

y(t – u)g ′′(u) du; y

)–

∫ μnνn y+ 3

2νn

y

(μn

νny +

32νn

– u)

g ′′(u) du.

Now taking the absolute value of both sides, we obtain

∣∣Q(μn ,νn)n,b (g; y) – g(y)

∣∣

≤∣∣∣∣Q(μn ,νn)

n,b

(∫ t

y(t – u)g ′′(u) du; y

)∣∣∣∣ +∣∣∣∣∫ μn

νn y+ 32νn

y

(μn

νny +

32νn

– u)

g ′′(u) du∣∣∣∣

≤Q(μn ,νn)n,b

(∣∣∣∣∫ t

y

∣∣(t – u)∣∣∣∣g ′′(u)

∣∣du∣∣∣∣; y

)+

∣∣∣∣(∫ μn

νn y+ 32νn

y

∣∣∣∣μn

νny +

32νn

– u∣∣∣∣∣∣g ′′(u)

∣∣du)∣∣∣∣

≤ ∥∥g ′′∥∥∞

{Q(μn ,νn)

n,b

(∣∣∣∣∫ t

y

∣∣(t – u)∣∣du

∣∣∣∣; y)

+∣∣∣∣(∫ μn

νn y+ 32νn

y

∣∣∣∣μn

νny +

32νn

– u∣∣∣∣du

)∣∣∣∣}

.

Therefore, by using the norm on g , we have

∣∣Q(μn ,νn)n,b (g; y) – g(y)

∣∣ ≤ ‖g‖C2B

{Q(μn ,νn)

n,b((t – y)2; y

)+

(μn

νny +

32νn

– y)2}

≤ ‖g‖C2B

{Q(μn ,νn)

n,b((t – y)2; y

)+

(Q(μn ,νn)

n,b (t – y; y))2}

≤ ‖g‖C2B

{ξμn ,νn (y) + χ2

μn ,νn (y)}

. (3.7)

Now, using the definition of auxiliary operators (3.5), we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤

∣∣∣∣Q(μn ,νn)n,b (f ; y) – f (y) + f

(μn

νny +

32νn

)– f (y)

∣∣∣∣


≤ ∣∣Q(μn ,νn)n,b (f – g; y)

∣∣ +∣∣Q(μn ,νn)

n,b (g; y) – g(y)∣∣

+∣∣g(y) – f (y)

∣∣ +∣∣∣∣f

(μn

νny +

32νn

)– f (y)

∣∣∣∣.

Combining (3.6) and (3.7) with the above equation, we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤ 3‖f – g‖∞ + ‖g‖C2

B

{ξμn ,νn (y) + χ2

μn ,νn (y)}

+ ‖f – g‖∞ + ω

(f ,

∣∣∣∣μn

νny +

32νn

– y∣∣∣∣)

≤ 4‖f – g‖∞ + ‖g‖C2B4ζμn ,νn (y) + ω

(f ,

∣∣χμn ,νn (y)∣∣),

and after taking the infimum on the right-hand side over all g ∈ C2B, we have

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤ 4K

(f , ζμn ,νn (y)

)+ ω

(f ,

∣∣χμn ,νn (y)∣∣)

≤ M(ω2

(f ,

√ζμn ,νn (y)

)+ min

{1, ζμn ,νn (y)

}‖f ‖∞)

+ ω(f ,

∣∣χμn ,νn (y)∣∣).

This completes the proof of the theorem. �

Theorem 3.6 Let f ∈ C1B[0,∞), then ∀y ≥ 0 and δ > 0,

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ {∣∣f ′(y)∣∣ + 2ω

(f ′, δn(y)

)}.

Proof Since f ∈ C1B[0,∞), we can write

f (t) – f (y) = f ′(y)(t – y) +∫ t

y

(f ′(u) – f ′(y)

)du. (3.8)

Now, using the well-known property of the modulus of continuity for δ > 0 and f ∈C1

B[0,∞),

∣∣f ′(u) – f ′(y)∣∣ ≤

( |u – y|δ

+ 1)

ω(f ′, δ

),

hence

∣∣∣∣∫ t

y

(f ′(u) – f ′(y)

)du

∣∣∣∣ ≤(

(t – y)2

δ+ |t – y|

)ω

(f ′, δ

)

Therefore, from (3.8) and the above equation, we have

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤ ∣∣f ′(y)

∣∣Q(μn ,νn)n,b

(|t – y|; y)

+(

1δQ(μn ,νn)

n,b((t – y)2; y

)+ Q(μn ,νn)

n,b(|t – y|; y

))ω

(f ′, δ

).


After applying the Cauchy–Schwarz inequality, we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤ (∣∣f ′(y)

∣∣ + ω(f ′, δ

))√Q(μn ,νn)

n,b((t – y)2; y

)√Q(μn ,νn)

n,b (1; y)

+(

1δQ(μn ,νn)

n,b((t – y)2; y

))ω

(f ′, δ

)

=(∣∣f ′(y)

∣∣ + ω(f ′, δ

))δn(y) +

(δ2

n(y)δ

)ω

(f ′, δ

).

Choosing δ = δn(y), we get our desired result. �

For f ∈ CB[0,∞), the Ditzian–Totik modulus of smoothness [4] of the first order is givenby

ωϕ(f , δ) = sup0<h≤δ

{∣∣∣∣f(

y +hϕ(y)

2

)– f

(y –

hϕ(y)2

)∣∣∣∣; y ± hϕ(y)2

∈ [0,∞)}

,

and an appropriate Peetre’s K-functional is defined by

Kϕ(f , δ) = infg∈Wϕ [0,∞)

{‖f – g‖∞ + δ‖ϕg ′‖∞}

, δ > 0,

where Wϕ[0,∞) := {g : g ∈ ACloc[0,∞),‖ϕg ′‖∞ < ∞} where g ∈ ACloc[0,∞) means g isabsolutely continuous on every compact subset [a, b] of [0,∞). It is known from [4] thatthere exists a constant M such that

M–1ωϕ(f , δ) ≤ Kϕ(f , δ) ≤ Mωϕ(f , δ). (3.9)

Now, we find the order of approximation of the sequence of operators (1.3) by means ofDitzian–Totik modulus of smoothness.

Theorem 3.7 For any f ∈ CB[0,∞) and y ∈ [0,∞),

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ Mωϕ

(f ,

δn(y)√y

).

Proof Let ϕ(y) = √y, then by Taylor’s theorem, for any g ∈ Wϕ[0,∞), we get

g(t) = g(y) +∫ t

yg ′(u) du = g(y) +

∫ t

y

g ′(u)ϕ(u)ϕ(u)

du,

therefore,

∣∣g(t) – g(y)∣∣ =

∥∥ϕg ′∥∥∞

∣∣∣∣∫ t

y

1ϕ(u) du

∣∣∣∣= 2

∥∥ϕg ′∥∥∞|√t –√

y|

= 2∥∥ϕg ′∥∥∞

|t – y|√t + √y

,


which gives

∣∣g(t) – g(y)∣∣ ≤ 2

∥∥ϕg ′∥∥∞|t – y|√y

= 2∥∥ϕg ′∥∥∞

|t – y|ϕ(y)

.

Using Lemma 2.1 and the above equation, for any g ∈ Wϕ[0,∞), we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ ∣∣Q(μn ,νn)n,b (f – g; y)

∣∣ +∣∣Q(μn ,νn)

n,b (g; y) – g(y)∣∣ +

∣∣g(y) – f (y)∣∣

≤ 2‖f – g‖∞ +2‖ϕg ′‖∞

ϕ(y)Q(μn ,νn)

n,b(|t – y|; y

).

Applying the Cauchy–Schwarz inequality yields

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ 2‖f – g‖∞ +2‖ϕg ′‖∞

ϕ(y)

√Q(μn ,νn)

n,b((t – y)2; y

)

= 2‖f – g‖∞ +2‖ϕg ′‖∞

ϕ(y)δn(y).

Taking infimum on the right-hand side over all g ∈ Wϕ[0,∞), we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤ 2Kϕ

(f ,

δn(y)√y

),

which leads to the required result with the help of the relation between Peetre’s K-functional and Ditzian–Totik modulus of smoothness as given by the relation (3.9). �

4 Approximation results in weighted spacesLet ν > 0. We denote Cν[0,∞) := {f ∈ C[0,∞) : |f (t)| ≤ Mf (1 + tν),∀t ≥ 0} equipped withthe norm

‖f ‖ν = supt∈[0,∞)

|f (t)|1 + tν

. (4.1)

Further, let C∗2 [0,∞) be the subspace of C2[0,∞) consisting of functions f such that

limt→∞ f (t)1+t2 exists.

Theorem 4.1 For each f ∈ C∗2 [0,∞) and r > 0, the following relation holds:

limn→∞ sup

x∈[0,∞)

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r = 0.

Proof Let y0 > 0 be arbitrary but fixed, then by (4.1), we can write

supy∈[0,∞)

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r

≤ supy≤y0

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r + supy>y0

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r


≤ supy≤y0

{∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣} + supy>y0

|Q(μn ,νn)n,b (f ; y) + f (y)|

(1 + y2)1+r .

Since |f (t)| ≤ ‖f ‖2(1 + y2), we get

supy∈[0,∞)

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r (4.2)

≤ ∥∥Q(μn ,νn)n,b (f ; y) – f (y)

∥∥C[0,y0] + ‖f ‖2 sup

y>y0

|Q(μn ,νn)n,b (1 + t2; y)|

(1 + y2)1+r + supy>y0

‖f (y)‖(1 + y2)1+r

= I1 + I2 + I3 (say). (4.3)

By Korovkin’s theorem, we can see that the sequence of operators {Q(μn ,νn)n,b (f ; y)} converges

uniformly to the function f on every closed interval [0, a] as n → ∞, (cf. [12, p. 149]).Therefore, for a given ε > 0, ∃n1 ∈N such that

I1 =∥∥Q(μn ,νn)

n,b (f ; y) – f (y)∥∥

C[0,y0] <ε

3, ∀n ≥ n1. (4.4)

By using Lemma 2.1, we can find n2 ∈N such that

∣∣Q(μn ,νn)n,b

(1 + t2; y

)–

(1 + y2)∣∣ <

ε

3‖f ‖2, ∀n ≥ n2,

or Q(μn ,νn)n,b

(1 + t2; y

)<

(1 + y2) +

ε

3‖f ‖2, ∀n ≥ n2.

Hence

I2 = ‖f ‖2 supy>y0

|Q(μn ,νn)n,b (1 + t2; y)|

(1 + y2)1+r

< ‖f ‖2 supy>y0

1(1 + y2)1+r

((1 + y2) +

ε

3‖f ‖2

)

< ‖f ‖2 supy>y0

(1

(1 + y2)r

)+

ε

3

<‖f ‖2

(1 + y20)r +

ε

3, ∀n ≥ n2. (4.5)

Now, using (4.1),

I3 = supy>y0

|f (y)|(1 + y2)1+r ≤ ‖f ‖2

(1 + y20)r . (4.6)

Let us denote n0 = max{n1, n2}, then by (4.4), (4.5), and (4.6), we get

I1 + I2 + I3 < 2‖f ‖2

(1 + y20)r +

2ε

3, ∀n ≥ n0. (4.7)

Choose y0 so large that

2‖f ‖2

(1 + y20)r <

ε

3. (4.8)


Then, combining (4.3), (4.7), and (4.8), we obtain

supy∈[0,∞)

|Q(μn ,νn)n,b (f ; y) – f (y)|

(1 + y2)1+r < ε, ∀n ≥ n0.

Hence, the proof is completed. �

Now, we will obtain the rate of convergence of the operators Q(μn ,νn)n,b (f ; y) defined by

(1.3) for the functions having derivatives of bounded variation. Let DBV[0,∞) be thespace of functions in C2[0,∞), which have the derivative of bounded variation on everyfinite subinterval of [0,∞). Here, we show at the point y, where f ′(y+) and f ′(y–) exist,the operators Q(μn ,νn)

n,b (f ; y) converge to the function f (y). A function f ∈ DBV[0,∞) canbe represented as

f (y) =∫ y

0g(t) dt + f (0),

where g denotes a function of bounded variation on every finite subinterval [0,∞). Manyresearchers studied in this direction and their work pertaining to this area is described inthe papers [2, 11, 19], etc.

In order to study the order of convergence of the operatorsQ(μn ,νn)n,b (f ; y) for the functions

having a derivative of bounded variation, we rewrite the operator (1.3) as follows:

Q(μn ,νn)n,b (f ; y) =

∫ ∞

0W (t, y)f (t) dt, (4.9)

where W (t, y) is a kernel given by

W (t, y) = νne–1(

1 –1b

)(b–1)μny ∞∑l=0

C(b)l (–(b – 1)μny)

l!χI(t),

χI(t) being the characteristic function of I = [ lνn

, l+1νn

].

Lemma 4.2 Let for all x > 0 and sufficiently large n,(1) λμn ,νn (t, y) =

∫ t0 W (u, y) du ≤ ξ2

μn ,νn (x)(y–t)2 , 0 ≤ t < y,

(2) 1 – λμn ,νn (t, y) =∫ ∞

t W (u, y) du ≤ ξ2μn ,νn (y)(t–y)2 , y ≤ t < ∞.

Proof Using Lemma 2.1 and the definition of the kernel, we get

λμn ,νn (t, y) =∫ t

0W (u, y) du

≤∫ t

0

(y – uy – t

)2

W (u, y) du

≤ 1(y – t)2

∫ t

0(u – y)2W (u, y) du.


Hence, we have

λμn ,νn (t, y) ≤ 1(y – t)2 Q

(μn ,νn)n,b

((u – y)2; y, b

)

≤ 1(y – t)2 ξ 2

μn ,νn (y).

In the same fashion, we can prove the other inequality, therefore, we omit the details. �

Let∨b

a f be the total variation of f on [a, b], i.e.,

b∨a

f = V(f ; [a, b]

)= sup

P∈P

( n∑i=1

∣∣f (yi) – f (yi–1)∣∣)

, (4.10)

where P is the set of all partitions P = {a = y0, y1, . . . , yn = b} of [a, b], whick also has theproperty

b∨a

f =c∨a

f +b∨c

f .

Let

f ′y (t) =

⎧⎪⎪⎨⎪⎪⎩

f ′(t) – f ′(y–), 0 ≤ t < y,

0, t = y,

f ′(t) – f ′(y+), y < t < ∞.

(4.11)

Theorem 4.3 Let f ∈ DBV[0,∞), y > 0, and n be sufficiently large, then we get

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣

≤∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣ψμn ,νn (y)

∣∣ +ξ 2μn ,νn (y)

y

[√

n]∑k=1

(y+y/k∨y–y/k

f ′y

)+

y√n

(y+y/k∨y–y/k

f ′y

)

+ξ 2μn ,νn (y)

y2

∣∣f (2y) – f (y) – yf ′(y+)∣∣ +

(Mf + |f (y)|

y2 + 4Mf

)ξ 2μn ,νn (y)

+∣∣f ′(y+)

∣∣∣∣ψμn ,νn (y)∣∣ +

∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣ξμn ,νn (y).

Proof By (4.11), we obtain

f ′(t) =12(f ′(y+) + f ′(y–)

)+ f ′

y (t) +12(f ′(y+) – f ′(y–)

)sgn(t – y)

+ δy(t)(

f ′(t) –12(f ′(y+) + f ′(y–)

)), (4.12)

where

δy(t) =

⎧⎨⎩

1, t = y,

0, t = y.


Now using Lemma 2.1, equations (4.9) and (4.12), we get

Q(μn ,νn)n,b (f ; y) – f (y)

=∫ ∞

0

(f (t) – f (y)

)W (t, y) dt

=∫ ∞

0

(∫ t

yf ′(u) du

)W (t, y) dt

=∫ ∞

0

[∫ t

y

{12(f ′(y+) + f ′(y–)

)+ f ′

y (u) +12(f ′(y+) – f ′(y–)

)sgn(u – y)

+ δy(u)(

f ′(u) –12(f ′(y+) + f ′(y–)

))}du

]W (t, y) dt.

Since∫ t

y δy(u) du = 0, we have

Q(μn ,νn)n,b (f ; y) – f (y)

=12(f ′(y+) + f ′(y–)

)∫ ∞

0(t – y)W (t, y) dt +

∫ ∞

0

(∫ t

yf ′y (u) du

)W (t, y) dt

+12(f ′(y+) – f ′(y–)

)∫ ∞

0|t – y|W (t, y) dt. (4.13)

Now, we break the second term on the right-hand side of the above equation as follows:

∫ ∞

0

(∫ t

yf ′y (u) du

)W (t, y) dt

= –∫ y

0

(∫ y

tf ′y (u) du

)W (t, y) dt +

∫ ∞

y

(∫ t

yf ′y (u) du

)W (t, y) dt

= –I1 + I2,

where

I1 =∫ y

0

(∫ y

tf ′y (u) du

)W (t, y) dt,

I2 =∫ ∞

y

(∫ t

yf ′y (u) du

)W (t, y) dt.

Taking the absolute value on both sides of (4.13), we have

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤

∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣Q(μn ,νn)

n,b (t – y; y)∣∣ + |I1| + |I2|

+∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣Q(μn ,νn)n,b

(|t – y|; y).


After applying the Cauchy–Schwarz inequality, we obtain

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣≤

∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣ψμn ,νn (y)

∣∣ + |I1| + |I2|

+∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣√Q(μn ,νn)

n,b((t – y)2; y

)

=∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣ψμn ,νn (y)

∣∣ + |I1| + |I2| +∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣ξμn ,νn (y). (4.14)

Now applying Lemma 4.2 and integration by parts, I1 can be written as

I1 =∫ y

0

(∫ y

tf ′y (u) du

)W (t, y) dt

=∫ y

0

(∫ y

tf ′y (u) du

)∂

∂tλμn ,νn (t, y) dt

=∫ y

0f ′y (t)λμn ,νn (t, y) dt.

On taking the absolute value of I1, we have

|I1| =∫ y

0

∣∣f ′y (t)

∣∣λμn ,νn (t, y) dt

≤∫ y–y/

√n

0

∣∣f ′y (t)

∣∣λμn ,νn (t, y) dt +∫ y

y–y/√

n

∣∣f ′y (t)

∣∣λμn ,νn (t, y) dt

= K1 + K2, say.

Since f ′y (y) = 0, by (4.11), we have

K1 =∫ y–y/

√n

0

∣∣f ′y (t) – f ′

y (y)∣∣λμn ,νn (t, y) dt.

Now, using Lemma 4.2,

K1 ≤ ξ 2μn ,νn (y)

∫ y–y/√

n

0

∣∣f ′y (t) – f ′

y (y)∣∣ dt(y – t)2 .

By the definition of total variation (4.10) and taking t = y – y/u, we obtain

K1 ≤ ξ 2μn ,νn (y)

∫ y–y/√

n

0

( y∨t

f ′y

)dt

(y – t)2

= ξ 2μn ,νn (y)

∫ √n

1

( y∨y–y/u

f ′y

)duy

.


Now, after breaking the integral into a sum, we have

K1 ≤ ξ 2μn ,νn (y)

y

[√

n]∑k=1

∫ k+1

k

( y∨y–y/u

f ′y

)du

≤ ξ 2μn ,νn (y)

y

[√

n]∑k=1

( y∨y–y/k

f ′y

)(∫ k+1

kdu

)

=ξ 2μn ,νn (y)

y

[√

n]∑k=1

( y∨y–y/k

f ′y

).

Since by Lemma 4.2, λμn ,νn (t, y) ≤ 1 and using (4.11), we get

K2 =∫ y

y–y/n

( y∨t

f ′y

)dt

≤( y∨

y–y/n

f ′y

)∫ y

y–y/ndt

=y√n

( y∨y–y/n

f ′y

).

Thus, we get

|I1| ≤ξ 2μn ,νn (y)

y

[√

n]∑k=1

( y∨y–y/k

f ′y

)+

y√n

( y∨y–y/n

f ′y

). (4.15)

Using Lemma 4.2, we can write

|I2| =∣∣∣∣∫ ∞

y

(∫ t

yf ′y (u) du

)W (t, y) dt

∣∣∣∣≤

∣∣∣∣∫ 2y

y

(∫ t

yf ′y (u) du

)∂

∂t(1 – λμn ,νn (t, y)

)dt

∣∣∣∣ +∣∣∣∣∫ ∞

2y

(∫ t

yf ′y (u) du

)W (t, y) dt

∣∣∣∣.

Now, applying integration by parts and (4.11), we get

|I2| =∣∣∣∣∫ 2y

yf ′y (u) du

(1 – λμn ,νn (2y, y)

)–

∫ 2y

yf ′y (t)

(1 – λμn ,νn (t, y)

)dt

∣∣∣∣+

∣∣∣∣∫ ∞

2y

(∫ t

y

(f ′(u) – f ′(y+)

)du

)W (t, y) dt

∣∣∣∣

≤∣∣∣∣∫ 2y

yf ′y (u) du

∣∣∣∣.ξ2μn ,νn (y)

y2 +∫ 2y

y

∣∣f ′y (t)

∣∣(1 – λμn ,νn )(t, y) dt

+∣∣∣∣∫ ∞

2y

(f (t) – f (y)

)W (t, y) dt

∣∣∣∣ +∣∣f ′(y+)

∣∣∣∣∣∣∫ ∞

2y(t – y)W (t, y) dt

∣∣∣∣= P1 + P2 + P3 + P4, say.


Now, by (4.11), we get

P1 =ξ 2μn ,νn (y)

y2

∣∣∣∣∫ 2y

yf ′y (u) du

∣∣∣∣

=ξ 2μn ,νn (y)

y2

∣∣∣∣∫ 2y

y

(f ′(u) – f ′(y+)

)du

∣∣∣∣


y2

∣∣f (2y) – f (y) – yf ′(y+)∣∣

and

P2 =∫ 2y

y

∣∣f ′y (t)

∣∣.(1 – λμn ,νn (t, y))

dt

=∫ y+y/

√n

y

∣∣f ′y (t)

∣∣.(1 – λμn ,νn (t, y))

dt

+∫ 2y

y+y/√

n

∣∣f ′y (t)

∣∣.(1 – λμn ,νn (t, y))

dt

= J1 + J2, say.

Using Lemma 4.2, 1 – λμn ,νn (t, y) ≤ 1 and (4.11), we get

J1 =∫ y+y/

√n

y

∣∣f ′y (t)

∣∣.(1 – λμn ,νn (t, y))

dt

≤∫ y+y/

√n

y

∣∣f ′y (t) – f ′

y (y)∣∣dt

≤∫ y+y/

√n

y

( t∨y

f ′y

)dt

≤(y+y/

√n∨

yf ′y

)∫ y+y/√

n

ydt

≤ y√n

(y+y/√

n∨y

f ′y

).

Now, again with the help of Lemma 4.2 and (4.11), we obtain

J2 =∫ 2y

y+y/√

n

∣∣f ′y (t)

∣∣(1 – λμn ,νn (t, y))

dt


∫ 2y

y+y/√

n

∣∣f ′y (t) – f ′

y (y)∣∣ dt(t – y)2 .

By using (4.10) and t = y + y/u, we get

J2 ≤ ξ 2μn ,νn (y)

∫ 2y

y+y/√

n

( t∨y

f ′y

)dt

(t – y)2


= ξ 2μn ,νn (y)

∫ √n

1

(y+y/u∨y

f ′y

)duy


y

[√

n]∑k=1

∫ k+1

k

(y+y/u∨y

f ′y

)du


y

[√

n]∑k=1

(y+y/k∨y

f ′y

)(∫ k+1

kdu

)

=ξ 2μn ,νn (y)

y

[√

n]∑k=1

(y+y/k∨y

f ′y

).

Hence, we derive

P2 ≤ y√n

(y+y/√

n∨y

f ′y

)+

ξ 2μn ,νn (y)

y

[√

n]∑k=1

(y+y/k∨y

f ′y

).

Now, we estimate P3. As t ≥ 2y, then using 2(t – y) ≥ t and t – y ≥ y, we get

P3 =∣∣∣∣∫ ∞

2y

(f (t) – f (y)

)W (t, y) dt

∣∣∣∣≤

∫ ∞

2y

∣∣f (t)∣∣W (t, y) dt +

∫ ∞

2y

∣∣f (y)∣∣W (t, y) dt

≤ Mf

∫ ∞

2y

(1 + t2)W (t, y) dt +

∣∣f (y)∣∣ ∫ ∞

2yW (t, y) dt

=(Mf +

∣∣f (y)∣∣) ∫ ∞

2yW (t, y) dt + Mf

∫ ∞

2yt2W (t, y) dt

≤ (Mf +

∣∣f (y)∣∣) ∫ ∞

2y

(t – y)2

y2 W (t, y) dt + Mf

∫ ∞

2y4(t – y)2W (t, y) dt

≤(

Mf + |f (y)|y2 + 4Mf

)∫ ∞

2y(t – y)2W (t, y) dt

≤(

Mf + |f (y)|y2 + 4Mf

)Q(μn ,νn)

n,b((t – y)2; y

)

=(

Mf + |f (y)|y2 + 4Mf

)ξ 2μn ,νn (y).

We can compute P4 as follows:

P4 =∣∣f ′(y+)

∣∣∣∣∣∣∫ ∞

2y(t – y)W (t, y) dt

∣∣∣∣≤ ∣∣f ′(y+)

∣∣∣∣∣∣∫ ∞

0(t – y)W (t, y) dt

∣∣∣∣=

∣∣f ′(y+)∣∣∣∣Q(μn ,νn)

n,b (t – y; y)∣∣

=∣∣f ′(y+)

∣∣∣∣ψμn ,νn (y)∣∣.


Hence, we get

|I2| ≤ξ 2μn ,νn (y)

y2

∣∣f (2y) – f (y) – yf ′(y+)∣∣

+y√n

(y+y/√

n∨y

f ′y

)+

ξ 2μn ,νn (y)

y

[√

n]∑k=1

(y+y/k∨y

f ′y

)

+(

Mf + |f (y)|y2 + 4Mf

)ξ 2μn ,νn (y) +

∣∣f ′(y+)∣∣∣∣ψμn ,νn (y)

∣∣. (4.16)

Now, from (4.14)–(4.16), we obtain

∣∣Q(μn ,νn)n,b (f ; y) – f (y)

∣∣ ≤∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣ψμn ,νn (y)

∣∣ + |I1| + |I2|

+∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣ξμn ,νn (y)

≤∣∣∣∣12(f ′(y+) + f ′(y–)

)∣∣∣∣∣∣ψμn ,νn (y)

∣∣

+ξ 2μn ,νn (y)

y

[√

n]∑k=1

( y∨y–y/k

f ′y

)+

y√n

( y∨y–y/

√n

f ′y

)

+ξ 2μn ,νn (y)

y2

∣∣f (2y) – f (y) – yf ′(y+)∣∣

+y√n

(y+y/√

n∨y

f ′y

)+

ξ 2μn ,νn (y)

y

[√

n]∑k=1

(y+y/k∨y

f ′y

)

+(

Mf + |f (y)|y2 + 4Mf

)ξ 2μn ,νn (y) +

∣∣f ′(y+)∣∣∣∣ψμn ,νn (y)

∣∣

+∣∣∣∣12(f ′(y+) – f ′(y–)

)∣∣∣∣ξμn ,νn (y),

which gives the desired result. �

5 Graphical examplesExample 5.1 Let us take f (x) = 5x4 – 11x3 + 2x2. The convergence of the sequence of oper-ators defined by Eq. (1.3) when μn = νn = n towards the function f (x) (cyan) is shown forn = 10, 50, 100, respectively, in Figs. 1–3 taking b = 2 (blue), b = 6 (black), and b = 15 (red).Figures 4–6 illustrate the convergence of the sequence of operators defined by Eq. (1.3)taking μn = n +

√n + 1, νn = n + 12 towards the function f (x) (cyan) for n = 10, 50, 100,

keeping the value of b the same.Also, a direct comparison between the convergence of the old operator applied to f

(when μn = νn = n discussed in [9]) (blue) and the new operator (red) defined in Eq. (1.3)towards f (x) (cyan) is shown in Figs. 7–9, respectively, for n = 10, 50, 100, and b = 10. Itis clear that the new operator exhibits faster convergence towards the limit than the oldoperator. Also, the new operator is giving flexibility in choosing parameters in the form ofthe sequences μn and νn.


Figure 1 Convergence of the operators when μn

= νn = n and n = 10


= νn = n and n = 50


= νn = n and n = 100


= n +√n + 1, νn = n + 12 and n = 10



= n +√n + 1, νn = n + 12 and n = 50


= n +√n + 1, νn = n + 12 and n = 100

Figure 7 Comparison between the operators whenμn = νn = n, b = 10 and n = 10




6 ConclusionsWe have modified the sequence of operators discussed in [9] and developed many ap-proximation properties such as direct theorems, rate of convergence in weighted spaces,and approximation for functions of bounded variation. Moreover, we have also shown theconvergence of old and modified new operators graphically.

AcknowledgementsThe author (K.J. Ansari) extends his appreciation to the Deanship of Scientific Research at King Khalid University forfunding this work through research groups program under grant number G.R.P-93-41.

FundingNot applicable.

Availability of data and materialsNot applicable.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors contributed equally and significantly in writing this paper. All authors read and approved the finalmanuscript.

Author details1Department of Mathematics, College of Science, King Khalid University, Saudi Arabia, Saudi Arabia. 2Department ofMedical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan. 3Departmentof Mathematics, Aligarh Muslim University, Aligarh, India. 4Department of Computer Science and InformationEngineering, Asia University, Taichung, Taiwan. 5Department of IT, Ibra College of Technology, Ibra, Sultanate of Oman.6Department of Computer Science, College of Computer Science, King Khalid University, Gregor, Abha, Saudi Arabia.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 12 February 2020 Accepted: 16 April 2020

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