Dunkl-Gamma Type Operators including Appell Polynomials

TL;DR

This paper introduces Dunkl-Gamma type operators constructed with Appell polynomials and explores their approximation capabilities, expanding the mathematical tools available for function approximation.

Contribution

It presents a novel class of Dunkl-Gamma operators incorporating Appell polynomials and analyzes their approximation properties.

Findings

Established convergence properties of the operators

Derived error estimates for approximation

Extended the operators to broader function classes

Abstract

The aim of the present paper is to introduce Dunkl-Gamma type operators in terms of Appell polynomials and to investigate approximating properties of these operators.

Equations

$$\left(T_{n}f\right)\left(x\right)=\frac{e^{-nx}}{g\left(1\right)}\sum _{k=0}^{\infty }{p}_k\left(nx\right)f\left(\frac{k}{n}\right)$$

$$\frac{a_k}{g\left(1\right)}\ge 0,k=0,1,2,...$$

$$g\left(t\right)e^{xt}=\sum _{k=0}^{\infty }{p}_k\left(x\right)t^k,$$

$$g\left(t\right)=\sum _{r=0}^{\infty }{a}_r{t}^r,a_0\ne 0$$

$$\left(P_{n}f\right)\left(x\right)=\frac{e^{-nx}}{g\left(1\right)}\sum _{k=0}^{\infty }{p}_k\left(nx\right)\frac{n^{\lambda +k+1}}{\Gamma \left(\lambda +k+1\right)}\underset{0}{\overset{\infty }{\int }}{e}^{-nt}{t}^{\lambda +k}f\left(t\right)dt$$

$$S_n^{\ast }\left(f;x\right)=\frac{1}{e_{\mu }\left(nx\right)}\sum _{k=0}^{\infty }\frac{{\left(nx\right)}^k}{{\gamma }_{\mu }\left(k\right)}f\left(\frac{k+2\mu {\theta }_k}{n}\right),n\in \mathbb{N}$$

$$e_{\mu }\left(x\right)=\sum _{k=0}^{\infty }\frac{x^k}{{\gamma }_{\mu }\left(k\right)},$$

$${\gamma }_{\mu }\left(2k\right)=\frac{2^{2k}k!}{\Gamma \left(k+\mu +\frac{1}{2}\right)}\text{ and }{\gamma }_{\mu }\left(2k+1\right)=\frac{2^{2k+1}k!\Gamma \left(k+\mu +\frac{3}{2}\right)}{\Gamma \left(\mu +\frac{1}{2}\right)}$$

$${\gamma }_{\mu }\left(k+1\right)=\left(k+1+2\mu {\theta }_{k+1}\right){\gamma }_{\mu }\left(k\right),k\in {\mathbb{N}}_0,$$

$$\theta_{k}=\left\{\begin{array}[c]{cc}0,&if\text{ }k=2p\\ 1,&if\text{ }k=2p+1\end{array}\right..$$

$$T_{\mu }\varphi \left(x\right)={\varphi }^{\prime }\left(x\right)+\mu \frac{\varphi \left(x\right)-\varphi \left(-x\right)}{x},$$

$$T_{\mu }{e}_{\mu }\left(xt\right)=t{e}_{\mu }\left(xt\right),$$

$$T_{\mu }{x}^n=\frac{{\gamma }_{\mu }\left(n\right)}{{\gamma }_{\mu }\left(n-1\right)}{x}^{n-1}.$$

$$T_{\mu }\left(fg\right)\left(x\right)=f\left(x\right)T_{\mu }g\left(x\right)+g\left(-x\right)T_{\mu }f\left(x\right)+f^{\prime }\left(x\right)\left[g\left(x\right)-g\left(-x\right)\right],$$

$$T_{\mu }\left(fg\right)\left(x\right)=f\left(x\right)T_{\mu }g\left(x\right)+g\left(x\right)T_{\mu }f\left(x\right).$$

$$D_n\left(f;x\right)=\frac{1}{e_{\mu }\left(nx\right)}\sum _{k=0}^{\infty }\frac{{\left(nx\right)}^k}{{\gamma }_{\mu }\left(k\right)}\frac{n^{k+2\mu {\theta }_k+\lambda +1}}{\Gamma \left(k+2\mu {\theta }_k+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{e}^{-nt}{t}^{{}^{k+2\mu {\theta }_k+\lambda }}f\left(t\right)dt$$

$$p_k\left(x\right)=\sum _{n=0}^k{\left(\genfrac{}{}{0ex}{}{k}{n}\right)}_{\mu }{a}_{k-n}{x}^n,{\left(a_k\right)}_{k\ge 0}$$

$$A\left(t\right)e_{\mu }\left(xt\right)=\sum _{k=0}^{\infty }\frac{p_k\left(x\right)}{{\gamma }_{\mu }\left(k\right)}{t}^k,$$

$$A\left(t\right)=\sum _{r=0}^{\infty }\frac{a_r}{{\gamma }_{\mu }\left(r\right)}{t}^r,a_0\ne 0$$

$${\left(\genfrac{}{}{0ex}{}{k}{n}\right)}_{\mu }=\frac{{\gamma }_{\mu }\left(k\right)}{{\gamma }_{\mu }\left(n\right){\gamma }_{\mu }\left(k-n\right)}.$$

$$D_n^{\ast }\left(f;x\right)=\frac{1}{e_{\mu }\left(nx\right)A\left(1\right)}\sum _{k=0}^{\infty }\frac{p_k\left(nx\right)}{{\gamma }_{\mu }\left(k\right)}\frac{n^{k+2\mu {\theta }_k+\lambda +1}}{\Gamma \left(k+2\mu {\theta }_k+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{e}^{-nt}{t}^{{}^{k+2\mu {\theta }_k+\lambda }}f\left(t\right)dt,$$

$$\sum _{k=0}^{\infty }\frac{p_k\left(nx\right)}{{\gamma }_{\mu }\left(k\right)}=A\left(1\right)e_{\mu }\left(nx\right),$$

$$\sum _{k=0}^{\infty }\frac{p_{k+1}\left(nx\right)}{{\gamma }_{\mu }\left(k\right)}=\left(nx\right)A\left(1\right)e_{\mu }\left(nx\right)$$

$$+\mu e_{\mu }\left(-nx\right)\left[A\left(1\right)-A\left(-1\right)\right]+A^{\prime }\left(1\right)e_{\mu }\left(nx\right)$$

$$\sum _{k=0}^{\infty }\frac{p_{k+2}\left(nx\right)}{{\gamma }_{\mu }\left(k\right)}=n^2{x}^{2}A\left(1\right)e_{\mu }\left(nx\right)+2nx{e}_{\mu }\left(nx\right)A^{\prime }\left(1\right)$$

$$+2\mu e_{\mu }\left(-nx\right)\left[A^{\prime }\left(1\right)-\frac{A\left(1\right)-A\left(-1\right)}{2}\right]$$

$$+A^{\prime \prime }\left(1\right)e_{\mu }\left(nx\right).$$

$$D_n^{\ast }\left(1;x\right)=1$$

$$D_n^{\ast }\left(t;x\right)=x+\frac{\mu }{n}\left\{\frac{e_{\mu }\left(-nx\right)}{e_{\mu }\left(nx\right)}\left[\frac{A\left(1\right)-A\left(-1\right)}{A\left(1\right)}\right]\right\}+\frac{1}{n}\left[\frac{A^{{}^{\prime }}\left(1\right)}{A\left(1\right)}+\lambda +1\right]$$

$$D_n^{\ast }\left(t^2;x\right)$$

$$+\frac{2\mu }{n^2}\frac{e_{\mu }\left(-nx\right)}{e_{\mu }\left(nx\right)}\frac{A^{{}^{\prime }}\left(1\right)+A^{{}^{\prime }}\left(-1\right)}{A\left(1\right)}+\frac{2{\mu }^2}{n^2}\left[\frac{A\left(1\right)-A\left(-1\right)}{A\left(1\right)}\right]$$

$$+\frac{\mu }{n^2}\left(2\lambda +3\right)\frac{e_{\mu }\left(-nx\right)}{e_{\mu }\left(nx\right)}\left[\frac{A\left(1\right)-A\left(-1\right)}{A\left(1\right)}\right]$$

$$+\frac{1}{n^2}\left[\frac{A^{\prime \prime }\left(1\right)}{A\left(1\right)}+\left(2\lambda +4\right)\frac{A^{\prime }\left(1\right)}{A\left(1\right)}+\left(\lambda +1\right)\left(\lambda +2\right)\right].$$

Full text

1. Introduction

Recently, linear positive operators constructed via generating functions and their further extentions are intensively studied by many research authors, for example, we refer the readers to [13, 14, 15, 16, 17, 19, 23, 24, 25]. In [14], Jakimovski et al. introduced linear positive operators in terms of Appell polynomials as follows

[TABLE]

under the assumption

[TABLE]

where $g\left(1\right)\neq 0.$ Here, Appell polynomials $p_{k}\left(x\right)$ are generated by

[TABLE]

where $g\left(t\right)$ is an analytic function in the disc $\left|t\right|<R~{}~{}(R>1)$

[TABLE]

(see [5]). In [6], Ciupa defined the following Durrmeyer type integral modification of the operators (1.1)

[TABLE]

under the assumption given by (1.2) where $\lambda\geq 0.$ Sucu [22] introduced Dunkl analogue of the Szasz operators by

[TABLE]

for any $x\in\left[0,\infty\right),$ $n\in\mathbb{N}$, $\mu\geq 0$ and $f\in C\left[0,\infty\right)$ by using Dunkl generalization of the exponential function $e_{\mu}\left(x\right)$ defined by [21]

[TABLE]

where the coefficients $\gamma_{\mu}$ are in the form

[TABLE]

for $k\in\mathbb{N}_{0},$ $\mu>-\frac{1}{2}.$ Moreover, the next recursion formula is satisfied

[TABLE]

where $\theta_{k}$ is

[TABLE]

Now, let us recall the Dunkl derivative operator [9, 10].

Let $\mu$ be a real number satisfying $\mu>-\frac{1}{2}.$ The Dunkl operator $T_{\mu}$ is defined by

[TABLE]

where $\phi\left(x\right)$ is an entire function. For $\mu=0,$ the operator $T_{\mu}$ gives the derivative operator. It is clear that

[TABLE]

[TABLE]

Moreover, the Dunkl generalization of the product of two function is given by

[TABLE]

which gives the next result if the function $g$ is an even function

[TABLE]

By the motivation this work, many authors studied Dunkl analogue of the several approximation operators for example, we refer the readers to [1, 4, 7, 11, 12, 20].

Wafi and Rao [26] constructed Dunkl analogue of Szasz Durrmeyer operators as

[TABLE]

for any $\lambda\geq 0,$ $x\in\left[0,\infty\right),$ $n\in\mathbb{N}$, $\mu\geq 0$ and $f\in C\left[0,\infty\right).$ The authors also examined pointwise approximation results in several functional spaces. They also studied weighted approximation results and gave rate of convergence for functions with derivative of bounded variation.

In [3], Ben Cheikh studied some properties of Dunkl-Appell $d-$ortogonal polynomials. In that work, Dunkl-Appell polynomials $p_{k}\left(x\right)$ defined by

[TABLE]

are generated by

[TABLE]

where $A\left(t\right)$ is an analytic function in the disc $\left|t\right|<R~{}~{}(R>1)$

[TABLE]

and Dunkl-binomial coefficient is defined by

[TABLE]

Note that $\gamma_{0}\left(k\right)=k!$ and $\binom{k}{n}_{0}=\binom{k}{n}.$

Inspired by the above works, for any $x\in\left[0,\infty\right),~{}f\in C\left[0,\infty\right),$ we introduce Dunkl analogue of the Appell Szasz Durrmeyer operators as

[TABLE]

where $\mu,\lambda\geq 0,~{}A\left(1\right)\neq 0,~{}~{}\frac{a_{k-n}}{A\left(1\right)}\geq 0,~{}\left(0\leq n\leq k\right),~{}~{}k=0,1,2,...,$ and $\gamma_{\mu}$ is defined by $\left(\ref{1b}\right)$ and $A\left(t\right)$ is given as in $\left(\ref{***}\right).$

Note that in the case of $\mu=0$ the operator $\left(\ref{appell-dunkl-s-d}\right)$ gives the operator $\left(\ref{2}\right)$, and for $A\left(t\right)=1$ the operator $\left(\ref{appell-dunkl-s-d}\right)$ reduces to the operator $\left(\ref{dunkl-s-d}\right).$

We organize the paper as follows. In section 2, we give some lemmas and obtain the convergence of the operators $\left(\ref{appell-dunkl-s-d}\right)$ with the help of universal Korovkin-type theorem. In section 3, we compute the rates of convergence of the operators $D_{n}^{\ast}\left(f\right)$ to $f$ by means of the usual and second modulus of continuity and Lipschitz class functions.

2. Approximation properties of the operators $D_{n}^{\ast}$

In what follows, we first give some lemmas and then prove the main theorem with the help of the well-known Korovkin Theorem.

Lemma 1**.**

From the generating function (1.10), the following equalities are satisfied

[TABLE]

[TABLE]

and

[TABLE]

Proof.

Taking $t\rightarrow 1,$ $x\rightarrow nx$ in (1.10), we get the first one. When we apply the Dunkl operator $T_{\mu}$ to both of sides of the equality $\left(\ref{dunkl-appell}\right)$, by using the relations (1.6), (1.7) and (1.8) we obtain the second and third relations. ∎

Lemma 2**.**

For the operators D_{n}^{\ast},\one can have

[TABLE]

[TABLE]

[TABLE]

Proof.

For $f\left(t\right)=1$ in the operator (1.12), we have

[TABLE]

from $\left(\ref{1}\right),$ it follows $D_{n}^{\ast}\left(1;x\right)=1.$ For $f\left(t\right)=t,$ the operator (1.12) reduces to

[TABLE]

By considering the equalities $\left(\ref{1}\right)$ and $\left(\ref{2.2}\right),$ we obtain

[TABLE]

Similarly, for $f\left(t\right)=t^{2},$ by means of the equalities (2.1), (2.2) and (2.3), it is seen that the equality (2.4) holds. ∎

Lemma 3**.**

For each $x\in\left[0,\infty\right),$ it follows from the results in Lemma 2

[TABLE]

[TABLE]

Theorem 1**.**

Let $D_{n}^{\ast}$ be the operators given by $\left(\ref{appell-dunkl-s-d}\right).$ Then, for any $f\in C\left[0,\infty\right)\cap E,$ the following relation holds

[TABLE]

uniformly on each compact subset of $\left[0,\infty\right),$ where

[TABLE]

Proof.

From the results in Lemma 2

[TABLE]

holds where the convergence holds uniformly in each compact subset of $\left[0,\infty\right).$ Then, applying the universal Korovkin type Theorem 4.1.4 (vi) given in [2] gives the desired result. ∎

3. Rates of Convergence

In this part, we calculate the order of approximation by means of the usual and second modulus of continuity and Lipschitz class functions. First of all, we recall some definitions as follows.

Let $f\in\widetilde{C}[0,\infty)$ and $\delta>0.$ The modulus of continuity of $f$ denoted by $\omega\left(f;\delta\right)$ is defined by

[TABLE]

where $\widetilde{C}[0,\infty)~{}$is the space of uniformly continuous functions on $[0,\infty).~{}$Then, for any $\delta>0$ and each $x\in[0,\infty)$, we have the following inequality

[TABLE]

Let $C_{B}\left[0,\infty\right)~{}$be the class of real valued functions defined on $\left[0,\infty\right)$ which are bounded and uniformly continuous with the norm $\left\|f\right\|_{C_{B}}=\sup_{x\in\left[0,\infty\right)}\left|f\left(x\right)\right|.$ The second modulus of continuity of $f\in C_{B}\left[0,\infty\right)$ is defined by

[TABLE]

Now, let us give the following definitions.

Definition 1**.**

Let $f$ be a real valued continuous function defined on $[0,\infty).$ Then $f$ is said to be Lipschitz continuous of order $\gamma$ on $[0,\infty)$ if

[TABLE]

for $x,y\in[0,\infty)$ with $M>0$ and $0<\gamma\leq 1.$ The set of Lipschitz continuous functions is denoted by Lip${}_{M}\left(\gamma\right).$

Definition 2**.**

[8]** Peetre’s $K$-functional of the function $f\in C_{B}\left[0,\infty\right)$ is defined by

[TABLE]

where

[TABLE]

and the norm

[TABLE]

It is clear that the following inequality

[TABLE]

holds for all $\delta>0.$ The constant $M$ is independent of $f$ and $\delta.$

Theorem 2**.**

For $f\in\widetilde{C}[0,\infty)\cap E,$ we have

[TABLE]

where $\Omega_{n}^{2}$ is given as in Lemma 3.

Proof.

From linearity and positivity of the operators $D_{n}^{\ast},$ by applying (3.1), we get

[TABLE]

From the Cauchy-Schwarz inequality for integration, one may write

[TABLE]

by using this inequality, it follows that

[TABLE]

If we now apply Cauchy-Schwarz inequality for sum on the right hand side of (3.5), we get

[TABLE]

where $\Omega_{n}^{2}\left(x\right)$ is as in the equality (2.5). When we consider (3.6) in (3.4), we obtain

[TABLE]

If we choose $\delta=\sqrt{\Omega_{n}^{2}\left(x\right)}$, we arrive at

[TABLE]

We note that $\Omega_{n}^{2}\left(x\right)$ goes to zero when $n\rightarrow\infty$. ∎

Theorem 3**.**

For $f\in Lip_{M}\left(\alpha\right),$ such that $0<\alpha\leq 1,$ $M\in\mathbb{R}^{+}$ we have

[TABLE]

where $\Omega_{n}^{2}$ is given in Lemma 3.

Proof.

Since $f\in Lip_{M}\left(\alpha\right),$ we can write from linearity

[TABLE]

By taking into account Lemma 3 and Hölder inequality, we get

[TABLE]

which ends the proof. ∎

Now, we give rate of convergence of the operators $D_{n}^{\ast}$ via Peetre’s K-functional.

Lemma 4**.**

For any $g\in C_{B}^{2}\left[0,\infty\right)$, we have

[TABLE]

where

[TABLE]

Proof.

From the Taylor’s series of the function $g\in C_{B}^{2}\left[0,\infty\right),$ we can write

[TABLE]

By operating by $D_{n}^{\ast}$ on both sides of this equality and then using the linearity of the operator, we get

[TABLE]

By considering Lemma 3, one can have

[TABLE]

So the proof is completed. ∎

Theorem 4**.**

For any $f\in C_{B}\left[0,\infty\right)$, we have

[TABLE]

where $M$ is a positive constant which is independent of $n$ and $\lambda_{n}\left(x\right)$ given by $\left(\ref{2.9}\right).$

Proof.

Let $g\in C_{B}^{2}\left[0,\infty\right).$ In view of Lemma 4, one can have

[TABLE]

from $\left(\ref{a11}\right)$ it follows

[TABLE]

By $\left(\ref{a14}\right)$, we obtain

[TABLE]

which completes the proof. ∎