On approximation process by certain modified Dunkl generalization of Szasz-Beta type operators

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Research Article

Online:ISSN 2008-949X

Journal Homepage:www.isr-publications.com/jmcs

On approximation process by certain modified Dunkl

gener-alization of Sz´asz-Beta type operators

Ç i ˘gdem Atakuta, Seda Karatekeb,∗, ˙Ibrahim B üy ükyazıcıa

a_{Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey.}

b_{Department of Mathematics and Computer Science, Faculty of Science and Letters, Istanbul Arel University, Istanbul, Turkey.}

Abstract

In this paper, we give a generalization of the Sz´asz-Beta type operators generated by Dunkl generalization of exponential function and obtain convergence properties of these operators by using Korovkin’s theorem and weighted Korovkin-type theo-rem. We also establish the order of convergence by using the modulus of smoothness and the weighted modulus of continuity.

Keywords:Dunkl type generalization, Genuine-Sz´asz beta operators, modulus of smoothness, Lipschitz functions.

2010 MSC:41A10, 41A25, 41A36.

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1. Introduction

Approximation theory is concerned with approximating functions of a given class using functions from another, usually more elementary, class. The theory of approximation of function is now an ex-tremely extensive branch of mathematical analysis and this theory has very important applications in other branches. As it is known, linear positive operators play an important role in the study of approxi-mation of functions. One of the best known of these operators is the Sz´asz operator introduced by Sz´asz [13] and it is as below: S_n(f; x) = e−nx ∞ X k=0 f k n (nx)k k! , x > 0, n ∈ N.

This operator is a generalization of Bernstein polynomials to the infinite interval. Sz´asz operators and their generalizations have been studied by many authors (see [1,4–10,12–15]). One of the generalizations of the Sz´asz operator including parameters anand bnwas given by ˙Ispir and Atakut [7] as follows:

San,bn_n (f; x) := Sn(f; an, bn; x) = e−anx ∞ X k=0 f k b_n (anx)k k! , x > 0, n ∈ N, (1.1) ∗_{Corresponding author}

Email addresses: atakut@science.ankara.edu.tr (Ç i ˘gdem Atakut), sedakarateke34@gmail.com (Seda Karateke), ibuyukyazici@gmail.com(˙Ibrahim B üy ükyazıcı)

doi:10.22436/jmcs.019.01.02

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where{an} , {bn} are given increasing and unbounded sequences of positive numbers such that lim n→_∞ 1 b_n =0, a_n b_n =1 + O 1 b_n . (1.2)

Note that the parameters an and bn have an important effect for a better approximation of the operator

San,bn_n . An example of this situation will be illustrated in following Figure1.

Figure 1: The effects of the anand bnparameters on the approximation (an= n2, bn= n2+1).

For ν, x ∈ [0,∞) and f ∈ C [0, ∞) , in [12], Sucu introduced the following generalization of the Sz´asz operators (later, it was called as Dunkl analogue of Sz´asz operators) by using the generalization of the exponential function eνgiven in [11]:

S∗_n(f; x) = 1 e_ν(nx) ∞ X k=0 f k + 2νθk n (nx)k γ_ν(k), where eν(x) = ∞ X r=0 xr γν(r) , (1.3)

and for r ∈N0and ν > −1₂, the coefficients γνare given by:

γ_ν(2r) = 2 2r_r_{!Γ r + ν +}1 2 Γ ν +1 2 , γν(2r + 1) = 22r+1r!Γ r + ν +3₂ Γ ν + 1 2 , (1.4)

where the function Γ is well-known gamma function and also for p ∈N, θr=

0, if r=2p, 1, if r=2p+1, the recursion relation

γν(r +1) = (2νθr+1+ r +1) γν(r) (1.5)

holds. Also, it is easily seen that if we select ν = 0, then the operator S∗_nturns into the operator Sn.

Very recently, in [1], C¸ ekim et al. have given the Dunkl analogue of Sz´asz-Beta type operators defined by Ln(f; x) = (n −1) eν(nx) ∞ X r=1 n + r − 2 r −1 (nx)r γν(r) ∞_Z 0 sr−1(1 + s)−n−r+1f (s) ds + f (0) eν(nx) , (1.6)

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where eν(x) and γν are as given in (1.3) and (1.4), respectively. In this paper, inspired by the operators

(1.1) and (1.6), for ν, x ∈ [0,∞) and f ∈ C [0, ∞) we define a modified Dunkl analogue of the Sz´asz-Beta type operators as follows:

Ln(f; an, bn, x) := L∗n(f; x) = (n −1) e_ν(anx) ∞ X r=1 n + r − 2 r −1 (anx)r γ_ν(r) ∞_Z 0 sr−1(1 + s)−n−r+1f s bn ds + f (0) e_ν(anx) . (1.7)

Here the sequences {an} and {bn} have the properties given in (1.2). Note that the well-known beta

function is defined by B (r, n − 1) = ∞ Z 0 sr−1(1 + s)−n−r+1ds. (1.8)

For m ∈N, the beta integral (1.8) can be written as

B (r + m, n − m − 1) = ∞ Z 0 sm+r−1(1 + s)−n−r+1ds. (1.9) 2. Main Results

In this section, we will give some important results for the operator L∗n. For the proofs of the next

theorems the following simple results are needed.

Lemma 2.1. Let f(t) = ti, i = 0, 1, 2, 3, 4. For the operator L∗ndefined by (1.7), the following statements hold:

L∗_n(1; x) = 1, (2.1) |L∗ n(t; x) − x| 6 1 n −2 a_n bn −1 x + 2ν n −2, (2.2) L∗_n t2; x − x26 1 (n −2) (n − 3) a2_n b2 n −1 x2+ 2an(1 + 2ν) b2 n(n −2) (n − 3) x + 4ν 2₊_6ν b2 n(n −2) (n − 3) , (2.3) L∗_n t3; x − x36 1 (n −2) (n − 3) (n − 4) a3 n b3 n − (n −2) (n − 3) (n − 4) x3 +6 (ν + 1)a 2 n b3 n x2+ 12ν 2₊_{20ν + 6 a} n b3 n x + 12ν 3₊_6ν2₊_20ν b3 n , (2.4) L∗_n t4; x − x46 1(n − 2) (n − 3) (n − 4) (n − 5) × a4 n b4 n − (n −2) (n − 3) (n − 4) (n − 5) x4 + (8ν + 12) a 3 n b4 n x3+ 24ν 2₊_{64ν + 36}_a2 n b4 n x2 + 32ν 3₊_112υ2₊_{96ν + 64}_a n b4 n x + 16ν 4₊_64ν3₊_84ν2₊_84ν b4 n . (2.5)

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Proof. Firstly, we consider f (t) = 1. Using (1.9) and definition of eν, we have L∗_n(1; x) = (n −1) e_ν(anx) ∞ X r=1 n + r − 2 r −1 (anx)r γ_ν(r) ∞_Z 0 sr−1(1 + s)−n−r+1ds + 1 e_ν(anx) = 1 e_ν(anx) ∞ X r=1 (anx)r γ_ν(r) + 1 e_ν(anx) = 1 e_ν(anx) ∞ X r=0 (anx)r γ_ν(r) =1.

Now, consider f (t) = t. For each n > 2, by using (1.5), (1.9), and eν(x), respectively, we obtain

L∗_n(t; x) = (n −1) eν(anx) ∞ X r=1 (anx)r γν(r) ∞ Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 s bn ds + f (0) eν(anx) = 1 b_ne_ν(anx) 1 n −2 ∞ X r=1 r(anx) r γ_ν(r) + f (0) e_ν(anx) = 1 b_ne_ν(anx) 1 n −2 ∞ X r=1 (anx)r γ_ν(r) (r +2νθr) − 1 b_ne_ν(anx) 2ν n −2 ∞ X r=1 (anx)r γ_ν(r)θr = 1 n −2 " an b_nx − n +2 − 2ν e_ν(anx) ∞ X r=1 (anx)r γ_ν(r) θr # . Thus, |L∗ n(t; x) − x| 6 1 n −2 " an b_nx − (n −2) x − 2ν e_ν(anx) ∞ X r=1 (anx)r γ_ν(r)θr # 6 1 (n −2) an b_n−1 x + 2ν (n −2) for all n > 2, which shows that (2.2) holds. For n > 3 and f (t) = t2 using (1.9), we get

L∗_n t2; x = (n −1) eν(anx) ∞ X r=1 n + r − 2 r −1 (anx)r γν(r) ∞_Z 0 sr−1(1 + s)−n−r+1 s 2 b2 n ds + f (0) eν(anx) = 1 b2 n(n −2) (n − 3) 1 eν(anx) ∞ X r=1 (anx)r γν(r) r (r +1) + f (0) eν(anx) . Then by using r (r +1) = − (r − 1) 2νθr+ (r +2νθr) (r −1) + 2r and (1.5), L∗_nt2; x = 1 b2 n(n −2) (n − 3) 1 eν(anx) _∞ X r=1 (anx)r γν(r) (r +2νθr) (r −1) − (r − 1) 2νθr+2r) = 1 b2_n(n −2) (n − 3) 1 eν(anx) _∞ X r=1 (anx)r γν(r) (r +2νθr) (r −1) − 2ν ∞ X r=1 (anx)r γν(r) (r −1) θr+ ∞ X r=1 (anx)r γν(r) 2r = 1 b2 n(n −2) (n − 3) 1 eν(anx) _∞ X r=0 (anx)r+1 γν(r +1) γν(r +1) γν(r) r −2ν ∞ X r=1 (anx)r γν(r) (r −1) θr+ ∞ X r=1 (anx)r γν(r) 2r .

Therefore, since θr6 1, we obtain L ∗ n t2; x− x2 6 1 (n −2) (n − 3) a2 n b2 n x2− (n −2) (n − 3) x2 +2νanx +2νanx +4ν2+2ν + 2anx +4ν

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= 1 (n −2) (n − 3) a2 n b2 n − (n −2) (n − 3) x2+(4νan+2an) b2 n x +4ν 2₊_6ν b2 n = 1 (n −2) (n − 3) a2_n b2 n −1 x2+ 4ν 2₊_6ν b2 n(n −2) (n − 3) .

Similarly, it can be shown that the inequalities (2.4) and (2.5) hold. The moments for the operator L∗nare stated in next lemma.

Lemma 2.2. The operators L∗nsatisfy the following inequalities:

∆₁:= L∗_n(t − x; x) 6 1 n −2 an bn −1 x + 2ν n −2, ∆₂:= L∗_n(t − x)2; x6 a2_n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2 + 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν n −2 x + 4ν 3₊_6ν b2 n(n −2) (n − 3) , (2.6) ∆₃:= L∗_n (t − x)4; x 6 a4_n (n −2) (n − 3) (n − 4) (n − 5) b4 n + 6a 2 n (n −2) (n − 3) b2 n − 4a 3 n (n −2) (n − 3) (n − 4) b3 n − 4an (n −2) bn +1 x4+ 8ν n −2− 24 (ν + 1) a2n b3 n(n −2) (n − 3) (n − 4) + (8ν + 12) a 3 n b4 n(n −2) (n − 3) (n − 4) (n − 5) + 12an(1 + 2ν) b2 n(n −2) (n − 3) x3 + 24ν 2₊_{64ν + 36 a}2 n b4 n(n −2) (n − 3) (n − 4) (n − 5) + 6 4ν 2₊_6ν b2 n(n −2) (n − 3) − 4 12ν 2₊_{20ν + 6 a} n b3 n(n −2) (n − 3) (n − 4) ! x2+ − 4 12ν 3₊_6ν2₊_20ν b3 n(n −2) (n − 3) (n − 4) + 32ν 3₊_112ν2₊_{96ν + 64}_a n b4 n(n −2) (n − 3) (n − 4) (n − 5) ! x + 16ν 4₊_64ν3₊_84ν2₊_84ν b4 n(n −2) (n − 3) (n − 4) (n − 5) . (2.7)

By using Lemma2.1 and applying the well-known Korovkin theorem, we have the following useful result.

Theorem 2.3. Let L∗_nbe given by (1.7). Then for any f ∈ C [0,∞) ∩ E, we have L∗_n(f; x) ⇒ f (x) as n → ∞

on each compact subset K of [0,∞). Here E =

f∈ C [0,∞) : f satisfies the condition lim

x→_∞

f (x) 1 + x2 <∞

.

The next theorem gives approximation of the operator L∗_nin the weighted space. Firstly, the concepts of weighted spaces are introduced. Let R+ _{= [0,}_{∞) and ρ(x) = 1 + x}2_{. The weighted spaces of the}

functions and the norm of Bρ(R+)are defined by,

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C_ρ(R+) =f∈ B_ρ(R+) : fis continuous onR+ , C∗_ρ(R+) = f∈ C_ρ(R+) : lim x→_∞ f (x) ρ(x) is finite , kfk_ρ= sup x∈[0,∞) f (x) ρ (x), f ∈ Bρ(R +₎ .

Note that a weighted Korovkin-type theorem is given by Gadjiev [2, 3]. The next theorem presents the approximation of the operators L∗n in the weighted space.

Theorem 2.4. For the operators L∗ngiven in (1.7) and each function f ∈ C∗ρ (R+), one has

lim

n→∞kL ∗

n(f; x) − f (x)kρ=0.

Proof. From (2.1), we can write limn→∞kL∗n(1; x) − 1kρ=0. By (2.2) and the following calculation

sup x∈[0,∞) |L∗ n(t; x) − x| 1 + x2 6 1 n −2 a_n b_n−1 sup x∈[0,∞) x 1 + x2+ 2ν n −2_x∈[0,sup_∞) x 1 + x2 6 1 n −2 a_n b_n −1 + 2ν n −2, we get lim n→_∞kL ∗ n(t; x) − xkρ=0.

Finally, by (2.3) and the following calculation sup x∈[0,∞) L∗_n t2; x − x2 1 + x2 6 1 (n −2) (n − 3) a2_n b2 n −1 sup x∈[0,∞) x2 1 + x2 + 2an(1 + 2ν) b2 n(n −2) (n − 3) sup x∈[0,∞) x 1 + x2 + 4ν2+6ν b2 n(n −2) (n − 3) sup x∈[0,∞) 1 1 + x2 6 1 (n −2) (n − 3) a2_n b2 n −1 + 2an(1 + 2γ) + 4ν 2₊_6ν b2 n(n −2) (n − 3) , we have lim n→_∞ L∗_n t2; x − x2 ρ =0.

Thus, we get limn→_∞kL∗n(f; x) − f (x)kρ = 0 for each f ∈ C∗ρ(R+)according to weighted Korovkin-type

theorem.

The simplest method of estimating the degree of approximation by positive linear operators is with the aid of the first and second order modulus of continuity given by:

ω(f; δ) = sup |x−y|6δ{|f(x) − f(y)| : x, y ∈ [0, ∞)} and ω₂(f; δ) = sup 0<h6δ {|f(x + h) − 2f(x) + f(x − h)| : x ∈ [0, ∞)} , respectively.

In the next theorems, we will give the degree of approximation using the operator L∗nby considering

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Theorem 2.5. Let f ∈ eC [0,∞) ∩ E. Then the operators given in (1.7) satisfy the following inequality |L∗

n(f; x) − f (x)| 6 2ω (f; δn,x),

where eC [0,∞) is the space of uniformly continuous functions on [0, ∞), ω is the first order modulus of continuity of f, and δn,x = s a2 n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2₊ 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν (n −2) x + 4ν 3₊_6ν b2 n(n −2) (n − 3) .

Furthermore, if f ∈ Lip_M(α), the following inequality |L∗

n(f; x) − f (x)| 6 K (τn(x))α/2

holds, where τn(x) = ∆2 given by (2.6).

Proof. In view of (2.4), one gets |L∗ n(f; x) − f (x)| 6 L∗n(|f (t) − f (x)| ; x) 6 1 +1 δL ∗ n(|t − x| ; x) ω (f_{; δ) 6} 1 +1 δ p ∆₂ ω (f; δ)

using Cauchy-Schwarz inequality. If we select δ = δn,x =

√

∆₂, we obtain the desired result. Thus, the proof of first part of the theorem is finished.

Now, let f ∈ Lip_M(α), then we have |L∗

n(f; x) − f (x)| 6 L∗n(|f (t) − f (x)| ; x) 6 KL∗n(|t − x| α

; x) . From Lemma2.2and using H ¨older’s inequality, one gets

|L∗

n(f; x) − f (x)| 6 K (∆2)α/2.

Then choosing τn(x) = ∆2, the proof of the theorem is now completed.

Lemma 2.6. Let CB[0,∞) be the space of continuous and bounded functions on [0, ∞) . For f ∈ C2B[0,∞) =

f∈ C_B[0,∞) : f0, f00 ∈ C_B[0,∞) , we have |L∗ n(f; x) − f (x)| 6 χn(x) kfk_C2 B[0,∞), where χ_n(x) = ∆₁+ ∆₂. Proof. For f ∈ C2

B[0,∞) , using the Taylor’s formula of the function f, we write

f (t) = f (x) + (t − x) f0(x) +(t − x) 2 2! f 00 (σ), σ ∈ (x, t) , by linearity of L∗_n, we get L∗_n(f; x) − f (x) = f0(x) L∗_n(t − x; x) + f 00 (σ) 2! L ∗ n (t − x)2; x= f0(x) ∆₁+f 00 (σ) 2! ∆2. Then using Lemma2.2, one obtains

|Ln(f; x) − f (x)| 6 1 n −2 a_n b_n−1 x + 2ν n −2 f 0 C2 B[0,∞)

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+1 2 a2_n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2 + 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν n −2 x + 4ν 3₊_6ν b2 n(n −2) (n − 3) f 00 C2_B[0,∞) 6 (∆1+ ∆2)kfkC2 B[0,∞).

Choosing χn(x) = ∆1+ ∆2 finishes the proof.

Theorem 2.7. The operators L∗nsatisfy the following inequality

|L∗ n(f; x) − f (x)| 6 2M min 1,χn(x) 2 kfk_CB[0,_∞)+ ω₂ f; r χn(x) 2 ! , where f ∈ CB[0,∞) , x ∈ [0, ∞), M is a positive constant independent of n and χn(x).

With the help of the Peetre’s K functional, and using the well-known relation between K2and ω2 one has

|L∗ n(f; x) − f (x)| 6 2M min 1,χn(x) 2 kfk_CB[0,∞)+ ω₂ f; r χ_n(x) 2 !

and the proof of the theorem is completed.

In the last theorem, we will give rate of approximation of the operators L∗_n in the weighted space by using the weighted modulus of continuity. The details about the weighted modulus of continuity are included below. For f ∈ C∗ρ(R+), the weighted modulus of continuity is defined by

Ω (f; δ) = sup

x∈[0,∞) |h|6δ

|f (x + h) − f (x)| (1 + h2_{) (1 + x}2₎

and has the following properties:

lim δ→0Ω (f; δ) = 0, and |f (s) − f (x)| 6 2 1 +|t − x| δ 1 + δ2 1 + x2 1 + (t − x)2 Ω (f; δ) , (2.8)

where t, x ∈ [0,∞). For further properties of the weighted modulus of continuity see [7]. Theorem 2.8. Let f ∈ C∗_ρ(R+). Then the following inequality holds:

sup x∈[0,∞) |L∗ n(f; x) − f (x)| (1 + x2₎3 6 Mν 1 + 1 n Ω f;√1 n , where Mνis a constant independent of n.

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Proof. By Lemma2.2and (2.8), we get |L∗ n(f; x) − f (x)| 6 (n −1) eν(anx) ∞ X r=1 (anx)r γν(r) ∞_Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 × f s b_n − f (x) ds + f (0) e_ν(anx) 6 2 1 + δ2 1 + x2 Ω (f; δ) (n −1) e_ν(anx) × ∞ X r=1 (anx)r γ_ν(r) ∞ Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1  1 + s bn − x δ   1 + s b_n − x 2! ds + f (0) eν(anx) =2 1 + δ2 1 + x2 Ω (f; δ) (n −1) e_ν(anx) ×    ∞ X r=0 (anx)r γ_ν(r) ∞_Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 ds +1 δ ∞ X r=0 (anx)r γ_ν(r) ∞ Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 s b_n − x ds + ∞ X r=0 (anx)r γ_ν(r) ∞_Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 s b_n − x 2 ds + 1 δ ∞ X r=0 (anx)r γν(r) ∞_Z 0 sr−1 (1 + s)n+r−1 n + r − 2 r −1 s bn − x 3 ds   . We use the Cauchy-Schwarz inequality to get

|Ln(f; x) − f (x)| 6 2 1 + δ2 1 + x2 Ω (f; δ) 1 +1 δ p ∆₂+ ∆₂+1 δ p ∆₂∆₃ . With the help of (2.6) and (2.7), we have

|L∗ n(f; x) − f (x)| 6 2 1 + δ2 1 + x2 Ω (f; δ) 1 + a2_n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2 + 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν n −2 x +1 δ s a2 n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2₊ 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν n −2 x +1 δ s a2 n (n −2) (n − 3) b2 n − 2an (n −2) bn +1 x2₊ 2an(1 + 2ν) b2 n(n −2) (n − 3) − 4ν n −2 x ∆₃

and finally, by choosing δ = √1

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References

[1] B. Ç ekim, Ü. Dinlemez Kantar, ˙I. Y üksel, Dunkl generalization of Szász beta-type operators, Math. Methods Appl. Sci., 40 (2017), 7697–7704. 1,1

[2] A. D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P P Korovkin, Dokl. Akad. Nauk SSSR., 218 (1974), 1001–1004. 2

[3] A. D. Gadjiev, Theorems of the type of P P Korovkin theorems, Math. Zametki, 20 (1976), 781–786. 2

[4] A. R. Gairola, Deepmala, L. N. Mishra, Rate of approximation by finite iterates of q-Durrmeyer operators, Proc. Nat. Acad. Sci. India Sect. A, 86 (2016), 229–234. 1

[5] A. R. Gairola, Deepmala, L. N. Mishra, On the q-derivatives of a certain linear positive operators, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 1409–1417.

[6] R. B. Gandhi, Deepmala, V. N. Mishra, Local and global results for modified Sz´asz–Mirakjan operators, Math. Method. Appl. Sci., 40 (2017), 2491–2504.

[7] N. ˙Ispir, C¸ . Atakut, Approximation by modified Szasz–Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 571–578. 1,2

[8] M. Le´sniewicz, L. Rempulska, Approximation by some operators of the Szasz–Mirakjan type in exponential weight spaces, Glas. Mat. Ser. III, 32 (1997), 57–69.

[9] V. N. Mishra, K. Khatri, L. N. Mishra, Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, J. Inequal. Appl., 2013 (2013), 11 pages.

[10] V. N. Mishra, S. Pandey, I. A. Khan, On a modification of Dunkl generalization of Sz´asz Operators via q-calculus, Eur. J. Pure Appl. Math., 10 (2017), 1067–1077. 1

[11] M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl., 73 (1994), 369–396. 1

[12] S. Sucu, Dunkl analogue of Sz´asz operators, Appl. Math. Comput., 244 (2014), 42–48. 1,1

[13] O. Sz´asz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Research Nat. Bur. Standards, 45 (1950), 239–245. 1

[14] S. Varma, S. Sucu, G. ˙Iç öz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl.,

64(2012), 121–127.

[15] Z. Walczak, On certain modified Szasz–Mirakjan operators for functions of two variables, Demonstratio Math., 33 (2000), 92–100. 1