Approximation properties of (p,q)-Meyer-Konig-Zeller Durrmeyer operators

TL;DR

This paper introduces (p,q)-Meyer-Konig-Zeller Durrmeyer operators, analyzes their convergence properties, and demonstrates their approximation capabilities through theoretical results and MATLAB simulations.

Contribution

It presents a novel (p,q)-based Durrmeyer modification of Meyer-Konig-Zeller operators with new convergence and approximation results.

Findings

Operators converge uniformly for continuous functions

Statistical approximation properties are established

Numerical examples confirm theoretical results

Abstract

In this paper, we introduce Durrmeyer type modification of Meyer-Konig-Zeller operators based on (p,q)-integers. Rate of convergence of these operators are explored with the help of Korovkin type theorems. We establish some direct results for proposed operators. We also obtain statistical approximation properties of operators. In last section, we show rate of convergence of (p,q)-Meyer-Konig-Zeller Durrmeyer operators for some functions by means of Matlab programming.

Equations

$$[n{]}_{p,q}=\frac{p^n-q^n}{p-q},n=1,\mathrm{2.....}$$

$$[n{]}_{p,q}!$$

$$\left[\begin{array}[]{c}n\\ k\end{array}\right]_{p,q}=\frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}.$$

$$(x+y{)}_{p,q}^n=\prod _{j=0}^{n-1}(p^{j}x+q^{j}y).$$

$${\beta }_{p,q}(t,s)={\int }_0^1{x}^{t-1}(1-qx{)}_{p,q}^{s-1}{d}_{p,q}x$$

$${\beta }_{p,q}(t,s)=p^{(s-1)(s-2)/2-(t-1)}{\beta }_{\frac{q}{p}}(t,s),$$

$${\beta }_{p,q}(t,s)=p^{((s+t-1)(s+t-2)-(t-1)(t-2))/2-t+1}\frac{[t-1]{!}_{p,q}[s-1]{!}_{p,q}}{[s+t-1]{!}_{p,q}}.$$

$$M_{n,p,q}(f;x)=\frac{1}{p^{\frac{n(n+1)}{2}}}\sum _{k=0}^{\infty }{\left[\begin{array}{c}n+k\\ k\end{array}\right]}_{p,q}{x}^k{p}^{-kn}(1-x{)}_{p,q}^{n+1}f\left(\frac{p^n[k{]}_{p,q}}{[n+k{]}_{p,q}}\right),x\in [0,1)$$

$$M_{n,p,q}(1;x)$$

$$M_{n,p,q}(t;x)$$

$$x^2\le M_{n,p,q}(t^2;x)$$

$${\tilde{M}}_{n,k}^{(p,q)}(f;x)=\frac{[n+1{]}_{p,q}}{p^n}\sum _{k=0}^{\infty }{m}_{n,k}^{(p,q)}(x)(pq{)}^{-}k{\int }_0^1{b}_{n,k}^{(p,q)}(qt)f(t)d_{p,q}t,0\le x<1,$$

$$m_{n,k}^{(p,q)}(x)=\frac{1}{p^{kn+n(n+1)/2}}{\left[\begin{array}{c}n+k\\ k\end{array}\right]}_{p,q}{x}^k(1-x{)}_{p,q}^{n+1},$$

$$b_{n,k}^{(p,q)}(qt)=\frac{1}{p^{k(n-1)+n(n-1)/2}}{\left[\begin{array}{c}n+k+1\\ k\end{array}\right]}_{p,q}(qt{)}^k(1-qt{)}_{p,q}^n$$

$${\int }_0^1{b}_{n,k}^{(p,q)}(qt)t^s{d}_{p,q}t=\frac{[n+k+1{]}_{p,q}![k+s{]}_{p,q}!}{[k{]}_{p,q}![n+k+s+1{]}_{p,q}!}\frac{(pq{)}^k}{[n+1{]}_{p,q}}{p}^{n(s+1)}$$

$$\sum\limits_{k=0}^{\infty}\left[\begin{array}[]{c}n+k\\ k\end{array}\right]_{p,q}\frac{x^{k}(1-x)_{p,q}^{n+1}}{[n+k]_{p,q}^{\underline{r}}}p^{(r-n)k}=\frac{\prod\limits_{j=0}^{r-1}(p^{n-j}-q^{n-j}x)}{[n]_{p,q}^{\underline{r}}}p^{\frac{(n-r)(n-r+1)}{2}}$$

$$\frac{1}{[n+k+r{]}_{p,q}}\le \frac{1}{q^r[n+k{]}_{p,q}},r\ge 0$$

$${\tilde{M}}_{n,k}^{(p,q)}\left(e_0;x\right)$$

$$\frac{x}{q^2}\left(1-\frac{q+1}{[n{]}_{p,q}}\right)\le {\tilde{M}}_{n,k}^{(p,q)}\left(e_1;x\right)$$

$${\tilde{M}}_{n,k}^{(p,q)}\left(e_2;x\right)$$

$${\tilde{M}}_{n,k}^{(p,q)}(e_1,x)$$

$$=$$

$${\tilde{M}}_{n,k}^{(p,q)}(e_2,x)$$

$$I_1=\frac{p(p+q)}{q^6}\frac{\left(p^n-q^{n}x\right)\left(p^{n-1}-q^{n-1}x\right)}{[n{]}_{p,q}[n-1{]}_{p,q}}.$$

$$I_2$$

$$I_3$$

$$+$$

$${\tilde{M}}_{n,k}^{(p,q)}\left({\psi }_1;x\right)$$

$${\tilde{M}}_{n,k}^{(p,q)}\left({\psi }_2;x\right)$$

$$K_2(f,\delta )=\underset{g\in W^2}{\inf }\{\Vert f-g\Vert +\delta \Vert g^{\prime \prime }\Vert \},$$

$$K_2(f,\delta )\le C{\omega }_2(f,\sqrt{\delta }),$$

$${\omega }_2(f,\sqrt{\delta })=\underset{0<h\le \sqrt{\delta }}{\sup }\underset{x,x+h\in [0,1]}{\sup }|f(x+2h)-2f(x+h)+f(x)|.$$

Full text

Approximation properties of $(p,q)-$Meyer-König-Zeller Durrmeyer operators **Honey SharmaA1, Cheena Gupta2, Ramapati Maurya3

** 1pro.sharma.h@gmail.com,

Department of Mathematics, Gulzar Group of Institutes,

Ludhiana, (Punjab) India

2guptacheena21@gmail.com

I K G Punjab Technical University,

Kapurthala, (Punjab) India

3ramapatimaurya@gmail.com

I K G Punjab Technical University,

Kapurthala, (Punjab) India

Department of Mathematics, Manav Rachna University,

Faridabad, (Haryana) India

1. Abstract

In this paper, we introduce Durrmeyer type modification of Meyer-König-Zeller operators based on $(p,q)-$integers. Rate of convergence of these operators are explored with the help of Korovkin type theorems. We establish some direct results for proposed operators. We also obtain statistical approximation properties of operators. In last section, we show rate of convergence of $(p,q)-$Meyer-König-Zeller Durrmeyer operators for some functions by means of Matlab programming.

Keywords: $q-$Calculus, $(p,q)-$Calculus, $(p,q)-$Meyer-König-Zeller operator, Durrmeyer type operators, Modulus of continuity, Peetre $K-$functional, Statistical convergence.

Mathematical subject classification: 41A25, 41A35.

2. Introduction and Preliminaries

Recently, Mursaleen et al. [10] introduced $(p,q)-$ analogue of Bernstein type operators. In the sequence, many researchers gave the $(p,q)-$analogue of various well known positive linear operators and study their approximation properties, for details one may refer to [1, 6, 9, 14, 16]. Now, We begin by recalling certain notations of $(p,q)$ calculus.

Let $0<q<p\leq 1$. The $(p,q)$- integer is defined as

[TABLE]

and the $(p,q)$-factorial is given by

[TABLE]

For integers $0\leq k\leq n$, the $(p,q)$-binomial coefficient is defined as

[TABLE]

Further, $(p,q)-$binomial function is expressed as

[TABLE]

Recently, Sharma [14] introduces the $(p,q)-$Beta function for $s,t\in\Re^{+}$ as

[TABLE]

and also obtain the relation between $(p,q)-$Beta function and $q-$Beta function as

[TABLE]

where, $\beta_{\frac{q}{p}}(t,s)$ is $\frac{q}{p}-$ analogue of beta function. Using $\beta_{q}(t,s)=\frac{[t-1]!_{q}[s-1]!_{q}}{[s+t-1]!_{q}}$ and $[n]!_{\frac{q}{p}}=p^{-n(n-1)/2}[n]!_{p,q}$, we can write

[TABLE]

For $p=1$, all the notations of $(p,q)-$ calculus are reduced to $q-$calculus and further details on $(p,q)-$ calculus can be found in [3, 11, 12]. .

In a recent studies, Kadak et al. [8] introduced a $(p,q)-$analogue of Meyer-König-Zeller operators, for $0<q<p\leq 1$, on a function defined on $[0,1]$ as

[TABLE]

and for $x=1,$ $M_{n,p,q}(f;1)=f(1)$.

Further, the moment of the operators are given in the following Lemma.

Lemma 1**.**

([8]) For all $x\in[0,1]$ and $0<q<p\leq 1$, we have

[TABLE]

In the past two decades, Studies of Durrmeyer variants of various operators remained the centre of attraction for the researchers, for which one may refer to [2, 5, 7, 15, 17]. Motivated by these studies, now we introduce the Meyer-König-Zeller Durrmeyer operators based on $(p,q)-$integers in the following section.

3. Construction of operator and Moment estimate

For $0<q<p\leq 1$ and function $f$ defined on $[0,1]$, the $(p,q)-$ Meyer-König-Zeller Durrmeyer operators are defined as follows:

[TABLE]

here,

[TABLE]

[TABLE]

and $\widetilde{M}_{n,k}^{(p,q)}(f;1)=1$. Before computing the moments of $(p,q)-$Meyer-König-Zeller Durrmeyer operators, we prove some lemmas as follows:

Lemma 2**.**

Let $0<q<p\leq 1$ and $s=0,1,2...$, we have

[TABLE]

Proof.

Lemma can be proved directly by using definition of $(p,q)-$beta operator and Equation (2.1). ∎

Lemma 3**.**

For $r=1,2\ldots$ and $n>r$, we have

[TABLE]

where $[n]_{p,q}^{\underline{r}}=[n]_{p,q}[n-1]_{p,q}[n-2]_{p,q}\ldots[n-r+1]_{p,q}$.

Lemma 4**.**

The identity

[TABLE]

holds.

Theorem 1**.**

For all $x\in[0,1]$, $n\in\mathbb{N}$ and $0<q<p\leq 1$, we have

[TABLE]

here, $e_{i}=t^{i}$ for $i=0,1,2$.

Proof.

First moment can be directly computed. We use the moments obtained for $(p,q)-$Meyer-König-Zeller operators in Lemma 1 to estimate moments of proposed Durrmeyer operators. By using Lemma 2 for $s=1$ and Lemma 3, we get lower bound of second moment as follows:

[TABLE]

By using inequality of Lemma 4, upper bound can be obtained as below:

[TABLE]

[TABLE]

To estimate third moment, we use Lemma 2 for $s=2$ and Lemma 3 as follows:

[TABLE]

By using lemma 4, $I_{1}$ can be obtained as

[TABLE]

Computations for $I_{2}$ are as follows:

[TABLE]

$I_{3}$ can be obtained as follows:

[TABLE]

[TABLE]

By using $I_{1}$, $I_{2}$ and $I_{3}$, we get upper bound of second moment. ∎

Corollary 2**.**

Central moments of operators are

[TABLE]

where $\psi_{i}(x)=(t-x)^{i}$ for $i=1,2$.

Proof.

By the linearity of $\widetilde{M}_{n,k}^{(p,q)}$ and Theorem 1, central moments can be obtained directly. ∎

Remark 1*.*

For $0<q<p\leq 1$, by simple computations $\lim_{n\rightarrow\infty}[n]_{p,q}=1/(p-q)$. In order to obtain results for order of convergence of the operator, we take $q_{n}\in(0,1)$, $p_{n}\in(q_{n},1]$ such that $\lim_{n\rightarrow\infty}p_{n}=\lim_{n\rightarrow\infty}q_{n}=1$, $\lim_{n\rightarrow\infty}p^{n}_{n}=a$ and $\lim_{n\rightarrow\infty}q^{n}_{n}=b$, so that $\lim_{n\rightarrow\infty}\frac{1}{[n]_{p_{n},q_{n}}}=0$. Such a sequence can always be constructed for example, we can take $q_{n}=1-1/2n$ and $p_{n}=1-1/3n$, clearly $\lim_{n\rightarrow\infty}p_{n}^{n}=e^{-1/3}$, $\lim_{n\rightarrow\infty}q_{n}^{n}=e^{-1/2}$ and $\lim_{n\rightarrow\infty}\frac{1}{[n]_{p_{n},q_{n}}}=0$.

4. Rate of convergence

We denote $W^{2}=\{g\in C[0,1]:g^{\prime},g^{\prime\prime}\in C[0,1]\}$. For $\delta>0,$   $K-$functional is defined as

[TABLE]

here norm $||.||$ denotes the supremum norm on $C[0,1]$. Following the well-known inequality given in DeVore and Lorentz [4], there exists an absolute constant $C>0$ such that

[TABLE]

here, $\omega_{2}(f,\sqrt{\delta})$ is second order modulus of continuity for $f\in C[0,1]$, defined as

[TABLE]

By $\omega(f,\delta)=\sup_{0<h\leq\delta}\sup_{x,x+h\in[0,1]}|f(x+h)-f(x)|$, we denote the usual modulus of continuity for $f\in C[0,1]$.

Theorem 3**.**

Let $(p_{n})_{n}$ and $(q_{n})_{n}$ be the sequence as defined in Remark 1. Then for each $f\in C[0,1]$, $\widetilde{M}_{n,k}^{(p_{n},q_{n})}(f;x)$ converges uniformly to $f$.

Proof.

By Korovkin theorem, it is sufficient to show that $\lim_{n\rightarrow\infty}\|\widetilde{M}_{n,k}^{(p_{n},q_{n})}(t^{m};x)-x^{m}\|=0$ for $m=0,1,2$. For $m=0$ results hold trivially. Using Theorem 1, we obtain the results for $m=1,2$ as follows:

[TABLE]

Finally,

[TABLE]

Hence the Theorem. ∎

Theorem 4**.**

Let $(p_{n})_{n}$ and $(q_{n})_{n}$ be the sequence as defined in Remark 1. Let $f\in C[0,1]$. Then for all $n\in N$, there exists an absolute constant $C>0$ such that

[TABLE]

here,

[TABLE]

and

[TABLE]

Proof.

For $x\in[0,1],$ we consider the operators $M_{n}^{*}(f;x)$ as

[TABLE]

Using first central moment of $\widetilde{M}_{n,k}^{(p_{n},q_{n})}$ and positivity of operator, we immediately get $M_{n}^{*}(t-x;x)=0$.

For $g\in W^{2}$ and $x\in[0,1]$, using the Taylor’s formula

[TABLE]

Therefore,

[TABLE]

Finally, we have

[TABLE]

Also, we have

[TABLE]

Therefore,

[TABLE]

On taking the infimum on the right hand side over all $g\in W^{2}$ and by the definition of $K-$functional, we get

[TABLE]

∎

5. Statistical Approximation

In this section, by using a Bohman-Korovkin type theorem proved in [13], we present the statistical approximation properties of purposed operator.

At this moment, we recall the concept of statistical convergence.

A sequence $(x_{n})_{n}$ is said to be statistically convergent to a number $L$, denoted by $st-\lim\limits_{n}x_{n}=L$ if, for every $\varepsilon>0$,

[TABLE]

where

[TABLE]

is the natural density of set $S\subseteq\mathbb{N}$ and $\chi_{S}$ is the characteristic function of S.

Let $C_{B}(D)$ represents the space of all continuous functions on D and bounded on entire real line, where D is any interval on real line. It can be easily shown that $C_{B}(D)$ is a Banach space with supreme norm. Also $\widetilde{M}_{n,k}^{(p,q)}(f;x)$ are well defined for any $f\in C_{B}([0,1])$.

Theorem 5**.**

([13]) Let $\left(L_{n}\right)_{n}$ be a sequence of positive linear operators from $C_{B}([a,b])$ into $B([a,b])$ and satisfies the condition that

[TABLE]

Then,

[TABLE]

Theorem 6**.**

Let $\{p_{n}\}_{n},\{q_{n}\}_{n}$ be sequences such that

[TABLE]

[TABLE]

Then, $\widetilde{M}_{n,k}^{(p_{n},q_{n})}(f,x)$ converges statistically to $f$. Therefore,

[TABLE]

Proof.

By Theorem 5, it is sufficient to prove that

[TABLE]

Based on Theorem 1, we have

[TABLE]

[TABLE]

and

[TABLE]

By taking supremum over $x\in[0,1]$ in above inequalities, we get

[TABLE]

and

[TABLE]

By using fact that $st-\lim_{n}q_{n}=1$ and $st-\lim_{n}p_{n}=1$, we get

[TABLE]

Hence the theorem. ∎

In the next theorem, we estimate the rate of convergence by using the concepts of modulus of continuity.

Theorem 7**.**

Let $\{p_{n}\}_{n},\{q_{n}\}_{n}$ be sequences such that

[TABLE]

[TABLE]

Then,

[TABLE]

for all $f\in C[0,1]$, here $\delta_{n}=\widetilde{M}_{n,k}^{(p_{n},q_{n})}((t-x)^{2};x)$.

Proof.

By the linearity and monotonicity of the operator, we get

[TABLE]

also, by property of modulus of continuity

[TABLE]

By using above facts, we get

[TABLE]

So, letting $\delta_{n}=\widetilde{M}_{n,k}^{(p_{n},q_{n})}\left((t-x)^{2};x\right)$ and take $\delta=\surd\delta_{n}$, we finally get result. ∎

6. Graphical Illustrations

In this section, we show approximation by $(p,q)-$Meyer-König-Zeller Kantrovich operators using Matlab programming for functions $f(x)=$ $(x-2/3)(x-4/5)$, $(x-1/4)(x-2/3)(x-4/5)$, $(x-1/3)(x-2/3)(x-3/5)(x-4/5)$ and $(x-1/3)(x-2/3)(x-3/5)(x-4/5)(x-5/7)$ taking $n=25$ and $k=150$.