A Dunkl Analogue of Operators Including Two-variable Hermite polynomials

TL;DR

This paper introduces a Dunkl generalization of operators involving two-variable Hermite polynomials and studies their approximation properties using classical tools like modulus of continuity and Peetre's K-functional.

Contribution

It presents a novel Dunkl analogue of operators with two-variable Hermite polynomials and analyzes their approximation behavior.

Findings

Operators effectively approximate functions with quantifiable error bounds.

Approximation properties are characterized using classical modulus of continuity.

Results extend classical Hermite polynomial operators through Dunkl generalization.

Abstract

The aim of this paper is to introduce a Dunkl generalization of the operators including two variable Hermite polynomials which are defined by Krech [14](Krech, G. A note on some positive linear operators associated with the Hermite polynomials, Carpathian J. Math., 32 (1) (2016), 71--77) and to investigate approximating properties for these operators by means of the classical modulus of continuity, second modulus of continuity and Peetre's K-functional.

Equations

$$S_{n}^{\ast}\left(g;x\right)=\frac{1}{e_{\nu}\left(nx\right)}\sum_{k=0}^{\infty}\frac{\left(nx\right)^{k}}{\gamma_{\nu}\left(k\right)}g\left(\frac{k+2\nu\theta_{k}}{n}\right)\ ;\ n\in\mathbb{N},~{}\nu,\ x\in[0,\infty)\$$

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

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

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

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

$$\sum _{n=0}^{\infty }\frac{H_n(\xi ,\alpha )}{n!}{t}^n=e^{\xi t+\alpha t^2}$$

$$H_n(\xi ,\alpha )=n!\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{{\alpha }^k{\xi }^{n-2k}}{k!(n-2k)!}.$$

$$G_n^{\alpha }\left(f;x\right)=e^{-\left(nx+\alpha x^2\right)}\sum _{k=0}^{\infty }\frac{x^k}{k!}{H}_k(n,\alpha )f\left(\frac{k}{n}\right),x\in {\mathbb{R}}_0^{+},n\in \mathbb{N}\text{ , }\alpha \ge 0$$

$$\sum _{n=0}^{\infty }\frac{H_n^{\mu }(\xi ,\alpha )}{n!}{t}^n=e^{\alpha t^2}{e}_{\mu }(\xi t)$$

$$H_n^{\mu }(\xi ,\alpha )=n!\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{{\alpha }^k{\xi }^{n-2k}}{k!{\gamma }_{\mu }(n-2k)},$$

$$h_n^{\mu }(\xi ,\alpha )=\frac{{\gamma }_{\mu }(n)H_n^{\mu }(\xi ,\alpha )}{n!}$$

$$\sum _{n=0}^{\infty }\frac{h_n^{\mu }(\xi ,\alpha )}{{\gamma }_{\mu }(n)}{t}^n=e^{\alpha t^2}{e}_{\mu }(\xi t)$$

$$h_n^{\mu }(\xi ,\alpha )={\gamma }_{\mu }(n)\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{{\alpha }^k{\xi }^{n-2k}}{k!{\gamma }_{\mu }(n-2k)}.$$

$${\mathbb{D}}_{\mu }(\phi (x))={\phi }^{{}^{\prime }}(x)+\frac{\mu }{x}(\phi (x)-\phi (-x)),x\in \mathbb{C}.$$

$$\begin{array}[]{cl}i)&\mathbb{D}_{\mu}^{j}:x^{n}\rightarrow\frac{\gamma_{\mu}(n)}{\gamma_{\mu}(n-j)}x^{n-j},j=0,1,2,...,n\ (n\in\mathbb{N)};~{}\mathbb{D}_{\mu}^{j}:1\rightarrow 0\\ &\\ ii)&\mathbb{D}_{\mu}(\varphi\psi)=\mathbb{D}_{\mu}(\varphi)\psi+\varphi\mathbb{D}_{\mu}(\psi),\ \text{where }\varphi\text{ is an even function}\\ &\\ iii)&\mathbb{D}_{\mu}:e_{\mu}(\lambda x)\rightarrow\lambda e_{\mu}(\lambda x).\end{array}$$

$$\begin{array}[]{cl}(i)&\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{h_{n+1}^{\mu}(\xi,\alpha)}{\gamma_{\mu}(n)}t^{n}=(\xi+2\alpha t)e^{\alpha t^{2}}e_{\mu}(\xi t)\\ (ii)&\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{h_{n+2}^{\mu}(\xi,\alpha)}{\gamma_{\mu}(n)}t^{n}=(\xi^{2}+4\xi\alpha t+4\alpha^{2}t^{2}+2\alpha)e^{\alpha t^{2}}e_{\mu}(\xi t)+4\alpha\mu e^{\alpha t^{2}}e_{\mu}(-\xi t)\end{array}$$

$$\begin{array}[]{l}\mathbb{D}_{\mu}(te_{\mu}(\xi t))=(t\xi+1)e_{\mu}(\xi t)+2\mu e_{\mu}(-\xi t)\\ \mathbb{D}_{\mu}(e^{\alpha t^{2}})=2\alpha te^{\alpha t^{2}}.\end{array}$$

$$\sum _{n=0}^{\infty }\frac{h_n^{\mu }(\xi ,\alpha )}{{\gamma }_{\mu }(n)}{\mathbb{D}}_{\mu }(t^n)={\mathbb{D}}_{\mu }(e^{\alpha t^2}{e}_{\mu }(\xi t)).$$

$$\sum _{n=0}^{\infty }\frac{h_{n+1}^{\mu }(\xi ,\alpha )}{{\gamma }_{\mu }(n)}{\mathbb{D}}_{\mu }(t^n)={\mathbb{D}}_{\mu }\left[(\xi +2\alpha t)e^{\alpha t^2}{e}_{\mu }(\xi t)\right]$$

$$\sum _{n=0}^{\infty }\frac{h_{n+2}^{\mu }(\xi ,\alpha )}{{\gamma }_{\mu }(n)}{t}^n=({\xi }^2+4\xi \alpha t+4{\alpha }^2{t}^2+2\alpha )e^{\alpha t^2}{e}_{\mu }(\xi t)+4\alpha \mu e^{\alpha t^2}{e}_{\mu }(-\xi t).$$

$$T_n(f;x):=\frac{1}{e^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^{k}f\left(\frac{k+2\mu {\theta }_k}{n}\right)$$

$$\begin{array}[]{cl}(i)&T_{n}(1;x)=1\\ (ii)&T_{n}(t;x)=x+\frac{2\alpha x^{2}}{n}\\ (iii)&T_{n}(t^{2};x)=x^{2}+\frac{4\alpha}{n^{2}}x^{2}+\frac{4\alpha}{n}x^{3}+\frac{4\alpha^{2}}{n^{2}}x^{4}+\frac{x}{n}+\frac{2\mu x}{n}\frac{e_{\mu}(-nx)}{e_{\mu}(nx)}\end{array}$$

$$T_n(t;x)=\frac{1}{n{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=1}^{\infty }\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k-1)}{x}^k.$$

$$T_n(t^2;x)=\frac{x}{n^2{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }(k+1+2\mu {\theta }_{k+1})\frac{h_{k+1}^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k.$$

$${\theta }_{k+1}={\theta }_k+(-1{)}^k,$$

$$T_n(t^2;x)$$

$${\Delta }_1$$

$${\Delta }_2$$

$$T_n\left(g;x\right)\stackrel{\text{uniformly}}{\rightrightarrows }g\left(x\right)$$

$$\left|g\left(s\right)-g\left(t\right)\right|\le M{\left|s-t\right|}^{\alpha }$$

Full text

1. Introduction

Up to now, linear positive operators and their approximation properties have been studied by many research workers, see for example [4], [5], [6], [9], [10], [19], [22] and references therein.** **Also, linear positive operators defined via generating functions and their further extensions are intensively studied by a large number of authors. For various extensions and further properties, we refer for example Altin et.al [1], Dogru et. al [8], Olgun et.al [17], Sucu et. al [21], Tasdelen et.al [23], Varma et.al [24, 25].

Recently, linear positive operators generated by a Dunkl generalization of the exponential function have been stated by many authors. In [20], Dunkl analogue of Szász operators by using Dunkl analogue of exponential function was given as follows

[TABLE]

for $g\in C[0,\infty)$ where Dunkl analogue of exponential function is defined by

[TABLE]

for $k\in\mathbb{N}_{0}$ and $\nu>-\frac{1}{2}$ and the coefficients $\gamma_{\nu}$ are as follows

[TABLE]

in [18]. Also, the coefficients $\gamma_{\nu}$ verify the recursion relation

[TABLE]

where

[TABLE]

for $p\in\mathbb{N}\cup\left\{0\right\}.$ Similarly, Stancu-type generalization of Dunkl analogue of Szász-Kantorovich operators and Dunkl generalization of Szász operators via q-calculus have been defined in [11, 12] and for other research see [15, 16].

The two-variable Hermite Kampe de Feriet polynomials $H_{n}(\xi,\alpha)$ are defined by (see [3])

[TABLE]

from which, it follows

[TABLE]

In a recent paper, Krech [14] has introduced the class of operators $G_{n}^{\alpha}$ given by

[TABLE]

in terms of two variable Hermite polynomials and investigated approximation properties of $G_{n}^{\alpha}$ .

In the present paper, we first give the Dunkl generalization of two variable Hermite polynomials and then we define a class of operators by using the Dunkl generalization of two variable Hermite polynomials. We give the rates of convergence of the operators $T_{n}$ to $f$ by means of the classical modulus of continuity, second modulus of continuity and Peetre’s $K$-functional and in terms of the elements of the Lipschitz class $Lip_{M}\left(\alpha\right).$

2. The Dunkl generalization of two variable Hermite polynomials

The Dunkl generalization of two variable Hermite polynomials is defined by

[TABLE]

from which, we conclude

[TABLE]

which gives the two variable Hermite polynomials as \mu=0.\For our purpose, we denote

[TABLE]

and we can write that the polynomials $h_{n}^{\mu}(\xi,\alpha)$ are generated by

[TABLE]

where

[TABLE]

In order to obtain some properties of $h_{n}^{\mu}(\xi,\alpha),$ we remind the following definition and lemma given in [18].

Definition 1**.**

[18]** Let ${\mu}\in\mathbb{C}_{0}~{}(\mathbb{C}_{0}:=\mathbb{C\setminus}\left\{-\frac{1}{2},-\frac{3}{2},...\right\},~{}x\in\mathbb{C}$ and let $\varphi$ be entire function. The linear operator $\mathbb{D}_{\mu}$ is defined on all entire functions $\varphi$ on $\mathbb{C}$ by

[TABLE]

We use the notation $\mathbb{D}_{\mu,x}$ since $\mathbb{D}_{\mu}$ is acting on functions of the variable $x$. Thus, $\mathbb{D}_{\mu,x}(\varphi(x))=\left(\mathbb{D}_{\mu}\varphi\right)(x).$

Lemma 1**.**

[18]** Let $\varphi,\psi$ be entire functions. For the linear operator $\mathbb{D}_{\mu}$, the following statements hold

[TABLE]

By using these definition and lemma, we can state the next result.

Lemma 2**.**

For the Dunkl generalization of two variable Hermite polynomials $h_{n}^{\mu}(\xi,\alpha)$, the following results hold true

[TABLE]

Proof.

Applying the linear operator $\mathbb{D}_{\mu}$ in view of Lemma 1 , we have

[TABLE]

Also applying the linear operator $\mathbb{D}_{\mu}$ to both side of the generating function (2.2), we have

[TABLE]

By using (2.4) and Lemma 1 (i), we get the first relation. Similarly, if we apply the linear operator $\mathbb{D}_{\mu}$ to the relation in (i), we get

[TABLE]

from (2.4) and Lemma 1, it follows

[TABLE]

Definition 2**.**

With the help of the Dunkl generalization of two variable Hermite polynomials given in (2.2), we introduce the operators $T_{n}(f;x),$ $n\in\mathbb{N}$ given by

[TABLE]

where $\alpha\geq 0,\mu\geq 0$ and $x\in\left[0,\infty\right).$ The operators (2.5) are linear and positive. In the case of $\mu=0,$ it gives $G_{n}^{\alpha}$ given by (1.6)

Lemma 3**.**

For the operators $T_{n}(f;x),$ we can obtain the following equations:

[TABLE]

Proof.

By using the generating function in (2.2), the relation $(i)$ holds. For the proof of $(ii),$ in view of the recursion relation in (1.4), we get

[TABLE]

When we replace $k$ by $k+1$, we obtain (ii) by use of Lemma 2 (i). For the proof of $(iii),$ by using (1.4), we have

[TABLE]

From the equation

[TABLE]

it yields

[TABLE]

Using the recursion relation in (1.4) in the first series, it follows

[TABLE]

from Lemma 2 (i) and (ii), we complete the proof of (iii).

Lemma 4**.**

As a consequence of Lemma 3, we can give the next results for $T_{n}$ operators

[TABLE]

Theorem 1**.**

For $T_{n}$ operators and any uniformly continuous bounded function $g$ on the interval $[0,\infty)$, we can give

[TABLE]

on each compact set $A\subset$ $[0,\infty)$ when $n\rightarrow\infty$.

Proof.

From Korovkin Theorem in [13], when n\rightarrow\infty,\we have $T_{n}\left(g;x\right)\overset{\text{uniformly}}{\rightrightarrows}g\left(x\right)$ on $A\subset[0,\infty)$ which is each compact set because $\lim_{n\rightarrow\infty}T_{n}(e_{i};x)=x^{i},$ for$\ i=0,1,2,$ which is uniformly on $A\subset[0,\infty)$ with the help of using Lemma 4.

Theorem 2**.**

The operator $T_{n}$ maps $C_{B}(\mathbb{R}_{0}^{+})$ into $C_{B}(\mathbb{R}_{0}^{+})$   and $\left\|T_{n}\left(f\right)\right\|\leq\left\|f\right\|$ for each $f\in C_{B}(\mathbb{R}_{0}^{+})$ .

3. Convergence of operators in (2.5)

In what follows we give some rates of convergence of the operators $T_{n}$. Firstly, we recall some definitions as follows. Let $Lip_{M}\left(\alpha\right)$ Lipschitz class of order $\alpha.$ If $g\in Lip_{M}\left(\alpha\right)$, the inequality

[TABLE]

holds where $s,t\in[0,\infty),\ 0<\alpha\leq 1$ and $M>0.$ $\widetilde{C}[0,\infty)$ is the space of uniformly continuous on $[0,\infty).$ The modulus of continuity $g\in\widetilde{C}[0,\infty)$ is denoted by

[TABLE]

We first estimate the rates of convergence of the operators $T_{n}$ by using modulus of continuity and in terms of the elements of the Lipschitz class $Lip_{M}\left(\alpha\right).$

Theorem 3**.**

If $h\in Lip_{M}\left(\alpha\right)$, we have

[TABLE]

where $\Delta_{2}$ is given in Lemma 4.

Proof.

Since $h\in Lip_{M}\left(\alpha\right)$, it follows from linearity

[TABLE]

From Lemma 4 and Hölder’s famous inequality, we can write

[TABLE]

Thus, we find the required inequality.

Theorem 4**.**

The operators in (2.5) verify the inequality

[TABLE]

where $g\in\widetilde{C}[0,\infty).$

Proof.

By Lemma 4, from Cauchy-Schwarz’s inequality and the property of modulus of continuity

[TABLE]

it follows

[TABLE]

Then from Lemma 4, one has

[TABLE]

by choosing $\delta=\frac{1}{\sqrt{n}}$, we completes the proof.

Let $C_{B}[0,\infty)$ denote the space of uniformly continuous and bounded functions on $[0,\infty)$. Also

[TABLE]

with the norm

[TABLE]

for $\forall g\in C_{B}^{2}[0,\infty).$

Lemma 5**.**

For $h\in C_{B}^{2}[0,\infty)$, the following inequality holds true

[TABLE]

where $\Delta_{1}$ and $\Delta_{2}$ are given by in Lemma 4.

Proof.

From the Taylor’s series of the function $h$,

[TABLE]

Applying the operator $T_{n}$ to both sides of this equality and then using the linearity of the operator, we have

[TABLE]

From Lemma 4, it yields

[TABLE]

which finishes the proof.

Now we recall that the second order of modulus continuity of $f$ on $C_{B}[0,\infty)$ is given as

[TABLE]

Peetre’s $K$-functional of the function $f\in C_{B}\left[0,\infty\right)$ is as follows

[TABLE]

The relation between $K$ and $\omega_{2}$ is as

[TABLE]

for all $\delta>0.$ Here $M$ is a positive constant. Now, we can give the important theorem.

Theorem 5**.**

For the operators by (2.5), the following inequality holds

[TABLE]

where $\forall g\in C_{B}[0,\infty),\ x\in[0,\infty)$, $M$ is a positive constant which is independent of $n$ and $\chi_{n}\left(x\right)=\Delta_{1}+\Delta_{2}$.

Proof.

For any $f\in C_{B}^{2}[0,\infty)$, from the triangle inequality, we can write

[TABLE]

from Lemma 5, which follows

[TABLE]

From (3.6), we have

[TABLE]

which holds

[TABLE]

from (3.7).

Similar to the proof of above theorem, simple computations give the next theorem.

Theorem 6**.**

If $g\in C_{B}[0,\infty)$ and $x\in[0,\infty)$, we get

[TABLE]

where $M$ is a positive constant.

Remark 1**.**

The case of $\mu=0$ in Theorem 6 gives the result given in [14].