A Dunkl-Gamma Type Operator in Terms of Two-Variable Hermite Polynomials

TL;DR

This paper introduces a new Dunkl-Gamma type operator utilizing two-variable Hermite polynomials and analyzes its approximation properties using classical tools like the modulus of continuity and Peetre's K-functional.

Contribution

It presents a novel Dunkl-Gamma operator based on two-variable Hermite polynomials and studies its approximation behavior with established mathematical tools.

Findings

The operator effectively approximates functions within the studied framework.

Approximation properties are characterized using classical modulus of continuity.

The operator's convergence is analyzed via Peetre's K-functional.

Abstract

The goal of this paper is to present a Dunkl-Gamma type operator with the help of two-variable Hermite polynomials and to derive its approximating properties via 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)\$$

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

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

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

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

$$D_n^f(x)=\frac{1}{e_{\mu }(nx)}\sum _{k=0}^{\infty }\frac{(nx{)}^k}{{\gamma }_{\mu }(k)}\frac{n^{k+2\mu {\theta }_k+\lambda +1}}{\Gamma (k+2\mu {\theta }_k+\lambda +1)}{\int }_0^{\infty }{t}^{k+2\mu {\theta }_k+\lambda }{e}^{-nt}f(t)dt,$$

$$\Gamma (x)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt\text{ for }x>0.$$

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

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

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

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

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

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

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

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

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

$${\mathcal{S}}_n(f;x):=\frac{1}{e^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k\frac{n^{k+2\mu {\theta }_k+\lambda +1}}{\Gamma (k+2\mu {\theta }_k+\lambda +1)}{\int }_0^{\infty }{t}^{k+2\mu {\theta }_k+\lambda }{e}^{-nt}f(t)dt$$

$$\begin{array}[c]{cl}(i)&\mathcal{S}_{n}(1;x)=1,\\ (ii)&\mathcal{S}_{n}(t;x)=x+\frac{2\alpha x^{2}}{n}+\frac{\lambda+1}{n},\\ (iii)&\begin{array}[c]{l}\mathcal{S}_{n}(t^{2};x)=\frac{x^{2}}{n^{2}}\left\{n^{2}+4n\alpha x+4\alpha^{2}x^{2}+2\alpha+4\alpha\mu\frac{e_{\mu}(-nx)}{e_{\mu}(nx)}\right\}\\ +\frac{2\mu x}{n^{2}}(n-2\alpha x)\frac{e_{\mu}(-nx)}{e_{\mu}(nx)}+\frac{2(\lambda+2)}{n^{2}}(n+2\alpha x)x+\frac{(\lambda+1)(\lambda+2)}{n^{2}}.\end{array}\end{array}$$

$${\int }_0^{\infty }{t}^{k+2\mu {\theta }_k+\lambda }{e}^{-nt}dt=\frac{\Gamma (k+2\mu {\theta }_k+\lambda +1)}{n^{k+2\mu {\theta }_k+\lambda +1}}.$$

$${\int }_0^{\infty }{t}^{k+2\mu {\theta }_k+\lambda +1}{e}^{-nt}dt=\frac{\Gamma (k+2\mu {\theta }_k+\lambda +2)}{n^{k+2\mu {\theta }_k+\lambda +2}}.$$

$${\mathcal{S}}_n(t;x)$$

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

$$=\frac{1}{n{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }(k+2\mu {\theta }_k)\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k+\frac{(\lambda +1)}{n{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k.$$

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

$${\int }_0^{\infty }{t}^{k+2\mu {\theta }_k+\lambda +2}{e}^{-nt}dt=\frac{\Gamma (k+2\mu {\theta }_k+\lambda +3)}{n^{k+2\mu {\theta }_k+\lambda +3}},$$

$${\mathcal{S}}_n(t^2;x)$$

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

$$+\frac{(\lambda +2)}{n^2{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }(k+2\mu {\theta }_k)\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k$$

$$+\frac{(\lambda +2)(\lambda +1)}{n^2{e}^{\alpha x^2}{e}_{\mu }(nx)}\sum _{k=0}^{\infty }\frac{h_k^{\mu }(n,\alpha )}{{\gamma }_{\mu }(k)}{x}^k.$$

$${\mathcal{S}}_n(t^2;x)$$

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

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

$${\mathcal{S}}_n(t^2;x)$$

Full text

1. Introduction

By now, several research workers have investigated linear positive operators and their approximation properties, see for instance [1], [2], [3], [4], [5], [6] and references so on.** **Furthermore, many authors have studied linear positive operators containing generating functions and given some approximation properties of these operators. To see such operators, we give the references such as Altın et. al [7], Doğru et. al [8], Krech [9], Olgun et. al [10], Sucu et. al [11], Taşdelen et. al [12], Varma et. al [13, 14].

Latterly, with the help of Dunkl exponential function, several authors have defined some linear positive operators. First of them is a Dunkl analogue of Szász operators given in [15] as follows:

[TABLE]

for $g\in C[0,\infty).$ Here the Dunkl exponential function is defined by

[TABLE]

for $\nu>-\frac{1}{2}$ and the coefficients $\gamma_{\nu}$ are given by

[TABLE]

Also, for the coefficients $\gamma_{\nu},$ the following recursion relation holds

[TABLE]

where $\theta_{k}$ is defined by

[TABLE]

for $p\in\mathbb{N}_{0}$ in [16]. Then, İçöz and Çekim have given a Stancu-type generalization of Szász-Kantorovich operators and $q-$Szász operators with the help of the Dunkl exponential function in [17, 18].

Next, Wafi and Rao [19] has introduced Szász–Gamma operators based on Dunkl analogue as

[TABLE]

where $\lambda\geq 0$ and $\Gamma$ is Gamma function defined by

[TABLE]

Finally, Aktaş et. al [20] has introduced the operator $T_{n}(f;x)$ for $n\in\mathbb{N}$

[TABLE]

where $\alpha\geq 0,\mu\geq 0$ and $x\in\left[0,\infty\right),$ via the Dunkl generalization of two-variable Hermite polynomials, $h_{n}^{\mu}(\xi,\alpha)$ in [21] defined as follows

[TABLE]

Here

[TABLE]

and $H_{n}^{\mu}(\xi,\alpha)$ has the following explicit representation

[TABLE]

We note that $H_{n}^{\mu}(\xi,\alpha)$ reduces to the two-variable Hermite polynomials defined by

[TABLE]

as \mu=0,\see detail [22]. In the case of $\mu=0,$ the operator $T_{n}(f;x)$ gives the operator $G_{n}^{\alpha}\left(f;x\right)$ defined by Krech [9] as follows

[TABLE]

where $H_{k}$ is the two variable Hermite polynomial in (1.9). Furthermore, recently, some sequences of Dunkl operators and Dunkl-Gamma type operators in terms of Appell polynomials have been defined and approximation properties of these operators have been investigated [23, 24].

The paper is organized as follows. In the next section, we introduce a Dunkl-Gamma type operator consisting of the generalization of two-variable Hermite polynomials. In the third section, the rates of convergence of the operator are obtained by means of the classical modulus of continuity, second modulus of continuity, Peetre’s $K$-functional and the Lipschitz class $Lip_{M}\left(\gamma\right).$

2. The Dunkl-Gamma Type Operator

Firstly, before we introduce our operator, let us give some features and results related to $h_{n}^{\mu}(\xi,\alpha)$ generated by the Dunkl generalization of two-variable Hermite polynomials in (1.8).

We first recall the following definition and lemma in [16].

Definition 1**.**

[16]** Assume that $\mu\in\mathbb{C},~{}x\in\mathbb{C}$. On all entire functions $\varphi$ on $\mathbb{C}$, Rosenblum defines the linear operator $\mathbb{D}_{\mu}$ as follows:

[TABLE]

Lemma 1**.**

[16]** Assume that $\varphi,\psi$ are entire functions. With the help of the linear operator $\mathbb{D}_{\mu}$, the following relations are satisfied:

[TABLE]

By using this definition and Lemma 1, the results in the next lemma hold true (see detail [20]).

Lemma 2**.**

[20]** $h_{n}^{\mu}(\xi,\alpha)$ has the following results

[TABLE]

Now we can define our operator as follows:

Definition 2**.**

Via $h_{n}^{\mu}(\xi,\alpha)$ given in (1.8), we consider the operator $\mathcal{S}_{n}(f;x),$ $n\in\mathbb{N}$ given by

[TABLE]

where$\ \alpha\geq 0,\ \mu>\frac{-1}{2},\ \lambda\geq 0$ and $x\in\left[0,\infty\right).$ We note that the operator in (2.2) is positive and linear. For $\alpha=0,$ it reduces to $D_{n}^{f}(x)$ given by (1.6).

Lemma 3**.**

The following equations can be derived from the definition of the operator $\mathcal{S}_{n}(f;x)$:

[TABLE]

Proof.

From the definition of Gamma function in (1.7), we have

[TABLE]

By using the above equation and the generating function in (1.8), we get the relation $(i)$. Using the definition of Gamma function again, we have

[TABLE]

Thus we get the relations

[TABLE]

The second series in right hand side of the above equation from the generating function in (1.8) is $\frac{(\lambda+1)}{n}.$ Also, if we use the recursion relation in (1.4) for the first term, we get

[TABLE]

While we are substituting $k$ by $k+1$ and using Lemma 2 $(i)$, we arrive at the relation $(ii)$. From the definition of Gamma function in (1.7) again, the following equality holds

[TABLE]

from which, it follows

[TABLE]

The third term in right hand side of the above equation from the generating function in (1.8) is $\frac{(\lambda+2)(\lambda+1)}{n^{2}}.$ Also by taking into account the recursion relation in (1.4) the for first and second series, we obtain

[TABLE]

Using the equation

[TABLE]

in [16], it yields

[TABLE]

Finally using the recursion relation in (1.4) in the first series, from Lemma 2 $(i)$ for the second and third series and Lemma 2 $(ii)$ for the first series, we complete the proof of $(iii)$. ∎

Remark 1**.**

In case of $\alpha=0,$ the results of Lemma 3 reduce to the results in the paper of Wafi and Rao in [19].

Lemma 4**.**

From the results of Lemma 3 and the linearity of the operator, we can obtain the next results for $\mathcal{S}_{n}$ operator

[TABLE]

Taking into account the inequality $\left|\tfrac{e_{\mu}(-x)}{e_{\mu}(x)}\right|\leq 1$ for $x\geq 0$ and $\mu>\frac{-1}{2}$ and $\tfrac{e_{\mu}(-x)}{e_{\mu}(x)}\rightarrow 0$ as $x\rightarrow\infty$ in [25], we have the following theorem.

Theorem 1**.**

Assume that the function $g$ on the interval $[0,\infty)$ is uniformly continuous bounded function. For each function $g$ on $[0,\infty),$ we can give

[TABLE]

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

Proof.

In view of Lemma 3

[TABLE]

is verified where the convergence holds uniformly in each compact subset of $[0,\infty)$. Then, using well known Korovkin Theorem in [26], we give the desired result. ∎

3. The Convergence Rates of Operator $\mathcal{S}_{n}$

In this part, we obtain some rates of convergence of the operator $\mathcal{S}_{n}$.

Theorem 2**.**

If $h\in Lip_{M}\left(\gamma\right)$, which satisfies the inequality

[TABLE]

where $s,t\in[0,\infty),\ 0<\gamma\leq 1$ and $M>0,$ we have

[TABLE]

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

Proof.

From the linearity of operator and $h\in Lip_{M}\left(\gamma\right),$ we get

[TABLE]

Under favour of Hölder’s inequality and Lemma 4, we can give the following required inequality

[TABLE]

∎

Theorem 3**.**

The operator $\mathcal{S}_{n}$ in (2.2) satisfies the inequality

[TABLE]

where $g\in\widetilde{C}[0,\infty),$ which is the space of uniformly continuous functions on [0,\infty),\and the modulus of continuity is defined by

[TABLE]

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

Proof.

Firstly we note that the modulus of continuity verifies the following inequality

[TABLE]

Under favour of the linearity of operator, Cauchy-Schwarz’s inequality, and Lemma 4, respectively, it follows

[TABLE]

By choosing $\delta=\frac{1}{\sqrt{n}}$, we complete the proof. ∎

Lemma 5**.**

For $h\in C_{B}^{2}[0,\infty)$, which is denoted by

[TABLE]

with the norm

[TABLE]

where $C_{B}[0,\infty)$ is the space of continuous and bounded functions on $[0,\infty)$ with the norm

[TABLE]

the following inequality holds true

[TABLE]

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

Proof.

With the help of the Taylor’s series of the function $h$, it follows that

[TABLE]

where $\varsigma$ between $x$ and s.\Then, by applying $\mathcal{S}_{n}$ to this equality and using the linearity of the operator, we get

[TABLE]

Using Lemma 4 and

[TABLE]

the following inequality is satisfied

[TABLE]

∎

For any $f\in C_{B}[0,\infty)$ and $\delta>0,$ Peetre’s $K$-functional is given by

[TABLE]

where $C_{B}^{2}[0,\infty)=\{g\in C_{B}[0,\infty):g^{\prime},g^{\prime\prime}\in C_{B}[0,\infty)\}$ and the second modulus of continuity $\omega_{2}\left(f;\delta\right)$ is defined as

[TABLE]

Also, the inequality holds

[TABLE]

between Peetre’s $K$-functional and second modulus of continuity $\omega_{2}$ (see [27]).

Lemma 6**.**

For $g\in C_{B}^{2}[0,\infty)$ and $x\geq 0,~{}\alpha,\lambda\geq 0,$ we get

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Because of the linearity property of the operator $\widetilde{\mathcal{S}}_{n}\left(g;x\right)=\mathcal{S}_{n}\left(g;x\right)+g(x)-g\left(x+\tfrac{2\alpha x^{2}+\lambda+1}{n}\right),$ for $t\in[0,\infty)$ we get $\widetilde{\mathcal{S}}_{n}\left(t-x;x\right)=0.$ For $g\in C_{B}^{2}[0,\infty)$, the Taylor’s expression is

[TABLE]

If we apply $\widetilde{\mathcal{S}}_{n}$ to the last equality and then use $\widetilde{\mathcal{S}}_{n}\left(t-x;x\right)=0,$ we have

[TABLE]

from which, it follows

[TABLE]

Since

[TABLE]

we have

[TABLE]

From (3.7), (3.8) and (3.9),

[TABLE]

where $\Upsilon(n,x)=\Lambda_{1}^{2}+\Lambda_{2}.$ ∎

Theorem 4**.**

Let $f\in C_{B}[0,\infty)$ and $c>0.$ The following inequality holds

[TABLE]

Proof.

For $f\in C_{B}[0,\infty)$ and $g\in C_{B}^{2}[0,\infty),$ we get

[TABLE]

On the other hand, we give

[TABLE]

Using Lemma 6, thus we have

[TABLE]

From the inequality (3.6) between Peetre’s $K$-functional and the second modulus of continuity $\omega_{2}$, we have the desired result. ∎

4. Data availability

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