Abstract
We construct the linear positive operators generated by the q-Dunkl generalization of the exponential function. We have approximation properties of the operators via a universal Korovkin-type theorem and a weighted Korovkin-type theorem. The rate of convergence of the operators for functions belonging to the Lipschitz class is presented. We obtain the rate of convergence by means of the classical, second order, and weighted modulus of continuity, respectively, as well.
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1 Introduction
In 1912, Bernstein [1] gave the following polynomials for any \(f\in C[0,1]\), \(x\in[0,1]\):
In 1950, for \(x\geq0\), Szász [2] introduced the operators
where \(f\in C[0,\infty)\).
q-calculus plays an important role in the natural sciences such as mathematics, physics, and chemistry. It has many applications in number theory, orthogonal polynomials, quantum theory, etc. There is a generalization of q-calculus, which is \((p,q)\)-calculus where \(0< q< p\leq1\). For \(p=1\), \((p,q)\)-integers reduce to q-integers. \((p,q)\)-integers are introduced to unify several forms of q-oscillator algebras in the representation theory of single parameter quantum algebras in physics. There have appeared some papers dealing with \((p,q)\)-calculus in recent years. Details are in [3].
We first mention some notations of q-calculus as found in [4, 5]. Let \(n\in \mathbb{N}_{0}\) and \(q\in(0,1)\). The q-integer \([ n ] _{q}\) and q-factorial \([ n ] _{q}!\) are, respectively, defined by
For \(n\in \mathbb{N}\), we have q-binomial coefficients
with \(\bigl[\scriptsize{ \begin{array}{@{}c@{}} n \\ 0 \end{array}} \bigr]_{q}=1\) and \(\bigl[\scriptsize{ \begin{array}{@{}c@{}} {n} \\ {k} \end{array}} \bigr]_{q}=0\) for \(k>n\). Then we give the following known representations:
q-Bernstein polynomials were first introduced by Lupaş [6] in 1987. A most useful definition of q-Bernstein polynomials was given by Phillips [7] as follows:
Many generalizations of q-Bernstein polynomials were given by authors such as Ostrovska [8], Büyükyazıcı [9, 10], Büyükyazıcı and Sharma [11], Aral [12], Nowak and Gupta [13], Gupta [14], Wang [15, 16], Wang and Wu [17], Phillips [18], Aral et al. [19], Acar and Aral [20], Aral and Gupta [21] and Finta and Gupta [22]. On the other hand, some authors dealt with generalizations of Szász-type operators [2, 12, 23–29].
Sucu [24] defined a Dunkl analog of Szász operators via a generalization of the exponential function given by [30] as
where \(\mu\geq0\), \(n\in \mathbb{N}\), \(x\geq0\), \(f\in C[0,\infty)\), and \(e_{\mu }(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{\gamma_{\mu}(n)}\). Here
and
There is a recursion relation for \(\gamma_{\mu}\),
where
(details are in [24]). İçöz and Çekim [25] investigated a Stancu-type generalization of a Kantorovich-type integral modification of the Dunkl analog of Szász operators by
where \(\mu\geq0\), \(n\in \mathbb{N}\), \(x\geq0\), \(\alpha,\beta\in \mathbb{R}\) (\(0\leq\alpha\leq\beta\)), and \(f\in C[0,\infty)\).
Ben Cheikh et al. [31] stated the q-Dunkl classical q-Hermite-type polynomials. They gave definitions of q-Dunkl analogs of exponential functions, recursion relations, and notations for \(\mu >-\frac{1}{2}\) and \(0< q<1\), respectively:
An explicit formula of \(\gamma_{\mu,q}(n)\) is
One can find some of the special cases \(\gamma_{\mu,q}(n)\) below:
Now, in this paper, we define a q-Dunkl analog of Szász operators as follows:
where \(\mu>\frac{1}{2}\), \(n\in \mathbb{N}\), \(x\geq0\), \(0< q<1\) and \(f\in C[0,\infty)\). Here \(e_{\mu,q}\) and \(\gamma _{\mu,q}\) are in (1.2), (1.5), respectively. Note that, when we take \(q\rightarrow1\), then we have (1.1).
2 Approximation properties
In this section, the convergence of the operators \(D_{n,q}\) is examined via a universal Korovkin-type theorem and a weighted approximation theorem given by [32].
Lemma 1
The operators \(D_{n,q}\) given by (1.6) satisfy the following:
Proof
For \(f(t)=1\), we have
One can easily see that
Using (2.6) and writing odd and even terms separately, we have
Using the inequality
it follows that
On the other hand, from (2.7), we have
So we have (2.3).
By the same computations, one gets (2.4) and (2.5). □
Lemma 2
The first, second, and fourth moments of the operators \(D_{n,q}\) are
Theorem 1
Let \(D_{n,q}\) be the operators given by (1.6). Then for any \(f\in C[0,\infty)\cap E\), the following relation:
holds uniformly on each compact subset of \([0,\infty)\), where
Proof
The proof is based on the well-known universal Korovkin-type theorem (see details in [33, 34]). □
We recall the weighted spaces of the functions which are defined on the positive semi-axis \(\mathbb{R} ^{+}=[0,\infty)\) as follows:
where \(\rho(x)=1+x^{2}\) is a weight function and \(M_{f}\) is a constant depending only on f. \(C_{\rho}(\mathbb{R}^{+})\) is a normed space with the norm \(\Vert f\Vert _{\rho }:=\sup_{x\geq0}\frac{\vert f(x)\vert }{\rho(x)}\).
Theorem 2
Let \(D_{n,q}\) be the operators given by (1.6). Then for any \(f\in C_{\rho}^{k}(\mathbb{R}^{+})\), we have
Proof
Using Lemma 1, one can easily prove the theorem. □
3 Rate of convergence
In this section, we compute the rate of convergence of the operators \(D_{n,q}\) with the help of Lipschitz class functions, and the classical, second order, and weighted modulus of continuity. For the sake of simplicity, we just give the theorems and lemmas without proofs in this section.
Lemma 3
Let \(f\in \operatorname{Lip}_{M}(\alpha)\) (\(0<\alpha\leq1\), \(M>0\)), i.e.
Then
holds where \(\vartheta_{n}(x)=D_{n,q}((t-x)^{2};x)\).
Theorem 3
Let \(f\in\widetilde{C}[0,\infty)\). Then the operators \(D_{n,q} \) verify
where \(\widetilde{C}[0,\infty)\) is the space of uniformly continuous functions on \([0,\infty)\), i.e. \(\omega ( f;\delta ) \) is the modulus of continuity of the function \(f\in\widetilde{C}[0,\infty )\) defined by
Lemma 4
Let \(g\in C_{B}^{2}[0,\infty)\). Then we get
where \(\vartheta_{n}(x)\) is given in Lemma 3 and \(C_{B}[0,\infty)\) is the space of all bounded and continuous functions on \([0,\infty)\) and
with the norm
Also
Theorem 4
For \(f\in C_{B}[0,\infty)\) and \(x\in[0,\infty)\), we get
where M is a positive constant and \(\omega_{2} ( f;\delta ) \) is the second order modulus of continuity of the function \(f\in C_{B}[0,\infty)\) defined as
and \(K_{2}(f;\delta)\) is the Peetre K-functional defined by
Theorem 5
Let \(f\in C_{\rho}^{k}( \mathbb{R}^{+})\). Then
holds. Here \(S_{\mu}\) is a constant independent of n.
4 Auxiliary results
In the section, we prove the theorems and lemmas given in the previous section.
Proof of Lemma 3
Since \(f\in \operatorname{Lip}_{M}(\alpha)\) and by linearity of the function f, we get
By using Lemma 1 and the Hölder inequality, one gets
This ends the proof. □
Proof of Theorem 3
From Lemma 1, the property of the modulus of continuity, and the Cauchy-Schwarz inequality, we have
If we choose \(\delta=\delta_{n}=\sqrt{\frac{1}{ [ n ] _{q}}}\), then we have desired result. □
Proof of Lemma 4
Using the generalized mean value theorem in the Taylor series expansion for \(g\in C_{B}^{2}[0,\infty)\), we have
By the linearity property of the operator \(D_{n,q}\), we obtain
From the above equality and Lemma 2, we conclude that
This ends the proof. □
Proof of Theorem 4
Let \(g\in C_{B}^{2}[0,\infty)\). From Lemma 4, we have
When we take the infimum over all \(g\in C_{B}^{2}[0,\infty)\), then we obtain
Now we recall the relation
where we have an absolute constant \(C>0\) [35], and we get (3.1). □
For arbitrary \(f\in C_{\rho}^{k}(\mathbb{R}^{+})\),the weighted modulus of continuity is defined by
and was introduced by Atakut and İspir in [23]. There are two main properties of this modulus of continuity, which are \(\lim_{\delta \rightarrow0}\Omega(f;\delta)=0\) and
where \(f\in C_{\rho}^{k}( \mathbb{R}^{+})\) and \(t,x\in[0,\infty)\). One can find many properties of the weighted modulus of continuity in [23].
Proof of Theorem 5
From Lemma 2 and (4.1), we have
Applying the Cauchy-Schwarz inequality for the above series, we obtain
From (2.9) and (2.10), we find
Choosing \(\delta=\frac{1}{\sqrt{[n]_{q}}}\) then the proof is completed. □
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İçöz, G., Çekim, B. Dunkl generalization of Szász operators via q-calculus. J Inequal Appl 2015, 284 (2015). https://doi.org/10.1186/s13660-015-0809-y
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DOI: https://doi.org/10.1186/s13660-015-0809-y