On modified Dunkl generalization of Szász operators via q-calculus

The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval [12,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\frac{1}{2},\infty)$\end{document} than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operators for functions belonging to the Lipschitz class.


Introduction and preliminaries
In , Bernstein  In the field of approximation theory, the application of q-calculus emerged as a new area. The first q-analogue of well-known Bernstein polynomials was introduced by Lupaş by applying the idea of q-integers []. In , Phillips [] considered another q-analogue of the classical Bernstein polynomials. Later on, many authors introduced q-generalizations of various operators and investigated several approximation properties [-].
We now present some basic definitions and notations of the q-calculus which are used in this paper [].

(.)
Our investigation is to construct a linear positive operator generated by a generalization of the exponential function defined by (see []) .
The recursion formula for γ μ is given by where μ > -  and Sucu [] defined a Dunkl analogue of Szász operators via a generalization of the exponential function [] as follows: Cheikh et al.
[] stated the q-Dunkl classical q-Hermite-type polynomials and gave definitions of q-Dunkl analogues of exponential functions and recursion relations for μ > -  and  < q < , Some of the special cases of γ μ,q (n) are defined as follows: In [], Içöz and Çekim gave the Dunkl generalization of Szász operators via q-calculus as follows: . Previous studies demonstrate that providing a better error estimation for positive linear operators plays an important role in approximation theory, which allows us to approximate much faster to the function being approximated.
Motivated essentially by Içöz and Çekim's [] recent investigation of Dunkl generalization of Szász-Mirakjan operators via q-calculus, we show that our modified operators have better error estimation than those in []. We also prove several approximation results and successfully extend the results of []. Several other related results are also discussed.

Construction of operators and moments estimation
Let Then, for any  n ≤ x <  -q n ,  < q < , μ >  n and n ∈ N, we define where e μ,q (x), γ μ,q are defined in (.), (.) by [] and f ∈ C ζ [, ∞) with ζ ≥  and Lemma . Let D * n,q (·; ·) be the operators given by (.). Then, for each  n ≤ x <  -q n , n ∈ N, we have the following identities/estimates: Now, by separating to the even and odd terms and using (.), we get On the other hand, we have This completes the proof.
Lemma . Let the operators D * n,q (·; ·) be given by Proof For the proof of this lemma, we use Lemma .. In view of This proves ().

Main results
We obtain the Korovkin-type approximation properties for our operators (see [-]).
Let C B (R + ) be the set of all bounded and continuous functions on R + = [, ∞), which is a linear normed space with Theorem . Let D * n,q (·; ·) be the operators defined by (.). Then, for any function f ∈ Proof The proof is based on Lemma . and the well-known Korovkin theorem regarding the convergence of a sequence of linear positive operators, so it is enough to prove the conditions This completes the proof.
We recall the weighted spaces of the functions on R + , which are defined as follows: where ρ(x) =  + x  is a weight function and M f is a constant depending only on f . Note that Q ρ (R + ) is a normed space with the norm f ρ = sup x≥ Then, for any function f ∈ Q k ρ (R + ), we have Theorem . Let D * n,q (·; ·) be the operators defined by (.). Then, for each function f ∈ Q k ρ (R + ), we have Proof From Lemma . and Theorem ., for τ = , the first condition is fulfilled. Therefore, Similarly, from Lemma . and Theorem ., for τ = ,  we have that which implies that This completes the proof.  ω(f , δ) gives the maximum oscillation of f in any interval of length not exceeding δ > , and it is given by

Rate of convergence
It is known that lim δ→+ ω(f , δ) =  for f ∈ C B [, ∞), and for any δ >  we have Now we calculate the rate of convergence of operators (.) by means of modulus of continuity and Lipschitz-type maximal functions.
Theorem . Let D * n,q (·; ·) be the operators defined by (.). Then, for f ∈ C B [, ∞), x ≥  n and n ∈ N, we have Proof We prove it by using (.), (.) and the Cauchy-Schwarz inequality. We can easily get if we choose δ = δ n,x , and by applying the result () of Lemma ., we get the result.
Remark . For the operators D n,q (·; ·) defined by (.) we may write that, for every f ∈ C B [, ∞), x ≥  and n ∈ N, where by [] we have Now we claim that the error estimation in Theorem . is better than that of (.) provided f ∈ C B [, ∞) and x ≥  n , n ∈ N. Indeed, for x ≥  n , μ ≥  n and n ∈ N, it is guaranteed that Now we give the rate of convergence of the operators D * n,q (f ; x) defined in (.) in terms of the elements of the usual Lipschitz class Lip M (ν).
Proof We prove it by using (.) and Hölder's inequality. We have Therefore, This completes the proof. Let with the norm Theorem . Let D * n,q (·; ·) be the operators defined in (.). Then for any g ∈ C  B (R + ) we have where δ n,x is given in Theorem ..
Proof Let g ∈ C  B (R + ). Then, by using the generalized mean value theorem in the Taylor series expansion, we have By applying the linearity property on D * n,q , we have which implies that The proof follows from () of Lemma ..
The Peetre's K -functional is defined by There exists a positive constant C >  such that K  (f , δ) ≤ Cω  (f , δ   ), δ > , where the second-order modulus of continuity is given by (.) Theorem . For x ≥  n , n ∈ N and f ∈ C B (R + ), we have where M is a positive constant, δ n,x is given in Theorem . and ω  (f ; δ) is the second-order modulus of continuity of the function f defined in (.).
Proof We prove this by using Theorem . From (.), clearly, we have g C B [,∞) ≤ g C  B [,∞) . Therefore, where δ n,x is given in Theorem .. By taking infimum over all g ∈ C  B (R + ) and by using (.), we get

Conclusion
The purpose of this paper is to provide a better error estimation of convergence by modification of the q-Dunkl analogue of Szász operators. Here we have defined a Dunkl generalization of these modified operators. This type of modification enables better error estimation on the interval [/, ∞) if compared to the classical Dunkl-Szász operators []. We obtained some approximation results via the well-known Korovkin-type theorem. We have also calculated the rate of convergence of operators by means of modulus of continuity and Lipschitz-type maximal functions.