A short note on approximation properties of Stancu generalization of q-Durrmeyer operators

Abstract

In the present paper, we introduce a simple Stancu generalization of q-analogue of well-known Durrmeyer operators. We first estimate moments of q-Durrmeyer-Stancu operators. We also establish the rate of convergence as well as Voronovskaja type asymptotic formula for q-Durrmeyer-Stancu operators.

1 Introduction

In the last decade, the applications of q-calculus in the approximation theory is one of the main areas of research. To approximate Lebesgue integrable functions on the interval $[0,1]$, Durrmeyer introduced the integral modification of the well-known Bernstein polynomials. In 1981, Derriennic [1] first studied these operators in detail. After the q-analogue of Bernstein polynomials by Phillips [2], Gupta and Heping [3] introduced q-Durrmeyer operators. Several other researchers have studied in this direction and obtained different approximation properties of many operators [4, 5]. In the present article, we propose the q-analogue of the Stancu generalization of Durrmeyer operators and study the convergence behavior. We have used notations of q-calculus as given in [6–8].

We set $p_{nk}(q;x)={\left(\begin{array}{c}n\\ k\end{array}\right)}_q{x}^k{(1-x)}_q^{n-k}$, $p_{\mathrm{\infty }k}(q;x)=\frac{x^k}{{(1-q)}^k{[k]}_q!}{(1-x)}_q^{\mathrm{\infty }}$.

Phillips [2] proposed the following q-Bernstein polynomials, which for each positive integer n and $f\in C[0,1]$ are defined as

$$B_{n,q}(f;x)=\sum _{k=0}^{n}f\left(\frac{{[k]}_q}{{[n]}_q}\right)p_{nk}(q;x).$$

In 2005, Derriennic [9] introduced a q-analogue of the Durrmeyer operators and has established some approximation properties of the q-Durrmeyer operators. After couple of years, Gupta [10] studied q-analogue of Durrmeyer operators and discussed approximation properties of the following q-Durrmeyer operators: For $f\in C[0,1]$,

$$\begin{array}{rl}{D}_{n,q}(f;x)& ={[n+1]}_q\sum _{k=0}^n{q}^{-k}{p}_{nk}(q;x){\int }_0^{1}f(t)p_{nk}(q;qt)d_{q}t\\ =\sum _{k=0}^n{A}_{nk}(f)p_{nk}(q;x);0\le x\le 1.\end{array}$$

(1)

Recently, Ibrahim et al. [11, 12] introduced Stancu generalization of certain operators and discussed its approximation properties. Motivated by such type operators, we introduce the Stancu type generalization of the q-Durrmeyer operators (1) for $0\le \alpha \le \beta$, which is defined as follows:

$$\begin{array}{rl}{D}_{n,q}^{\alpha ,\beta }(f;x)& ={[n+1]}_q\sum _{k=0}^n{q}^{-k}{p}_{nk}(q;x){\int }_0^{1}f\left(\frac{{[n]}_{q}t+\alpha }{{[n]}_q+\beta }\right)p_{nk}(q;qt)d_{q}t\\ =\sum _{k=0}^n{A}_{nk}^{\alpha ,\beta }(f)p_{nk}(q;x);0\le x\le 1.\end{array}$$

(2)

It can be easily verified that in case $q=1$, $\alpha =0$ and $\beta =0$, the operators defined in (2) reduce to the well-known Durrmeyer operators as defined in [1]. Throughout the present manuscript, the expression $g_n(x)\Uparrow g(x)$ means uniform convergence of a sequence $\{g_n(x)\}$ to $g(x)$.

The present note deals with the study of q-Durrmeyer-Stancu operators $\{D_{n,q}^{\alpha ,\beta }(f)\}$ for $0<q<1$. First, we estimate the moments for q-Durrmeyer-Stancu operators. We also study the rate of convergence as well as asymptotic formula for these operators $\{D_{n,q}^{\alpha ,\beta }(f)\}$. We establish a direct results in terms of $\omega (f,\cdot )$.

2 Estimation of moments

In this section, we shall obtain $D_{n,q}^{\alpha ,\beta }(t^i;x)$, $i=0,1,\dots$ .

Note that for $s=0,1,\dots$ and by the definition of q-Beta function [7], we have

$$\begin{array}{r}{\int }_0^1{t}^s{p}_{nk}(q;qt)d_{q}t=\frac{q^k{[n]}_q!{[k+s]}_q!}{{[n+s+1]}_q!{[k]}_q!}\text{and}\\ {\int }_0^1{t}^s{p}_{\mathrm{\infty }k}(q;qt)d_{q}t={(1-q)}^{s+1}\frac{q^k{[k+s]}_q!}{{[k]}_q!}.\end{array}$$

(3)

Lemma 1 We have

$$\begin{array}{c}{D}_{n,q}^{\alpha ,\beta }(1;x)=1,\hfill \\ D_{n,q}^{\alpha ,\beta }(t;x)=\frac{{[n]}_q+\alpha {[n+2]}_q+qx{[n]}_q^2}{{[n+2]}_q({[n]}_q+\beta )},\hfill \\ D_{n,q}^{\alpha ,\beta }(t^2;x)=\frac{q^3{[n]}_q^3({[n]}_q-1)x^2+((q{(1+q)}^2+2\alpha q^4){[n]}_q^3+2\alpha q{[3]}_q{[n]}_q^2)x}{{({[n]}_q+\beta )}^2{[n+2]}_q{[n+3]}_q}\hfill \\ \phantom{D_{n,q}^{\alpha ,\beta }(t^2;x)=}+\frac{{\alpha }^2}{{({[n]}_q+\beta )}^2}+\frac{(1+q+2\alpha q^3){[n]}_q^2+2\alpha {[3]}_q{[n]}_q}{{({[n]}_q+\beta )}^2{[n+2]}_q{[n+3]}_q}.\hfill \end{array}$$

Lemma 2 We have

$$\begin{array}{c}{\delta }_n(x)=D_{n,q}^{\alpha ,\beta }(t-x,x)=(\frac{q{[n]}_q^2}{{[n+2]}_q({[n]}_q+\beta )}-1)x+\frac{{[n]}_q+\alpha {[n+2]}_q}{{[n+2]}_q({[n]}_q+\beta )},\hfill \\ {\gamma }_n(x)=D_{n,q}^{\alpha ,\beta }({(t-x)}^2,x)\hfill \\ \phantom{{\gamma }_n(x)}=\frac{q^4{[n]}_q^4-q^3{[n]}_q^3-2q{[n]}_q^2{[n+3]}_q({[n]}_q+\beta )+{[n+2]}_q{[n+3]}_q{({[n]}_q+\beta )}^2}{{({[n]}_q+\beta )}^2{[n+2]}_q{[n+3]}_q}{x}^2\hfill \\ \phantom{{\gamma }_n(x)=}+\frac{q{(1+q)}^2{[n]}_q^3+2q\alpha {[n]}_q^2{[n+3]}_q-(2{[n]}_q+2\alpha {[n+2]}_q){[n+3]}_q({[n]}_q+\beta )}{{({[n]}_q+\beta )}^2{[n+2]}_q{[n+3]}_q}x\hfill \\ \phantom{{\gamma }_n(x)=}+\frac{(1+q){[n]}_q^2+2\alpha {[n]}_q{[n+3]}_q}{{({[n]}_q+\beta )}^2{[n+2]}_q{[n+3]}_q}.\hfill \end{array}$$

Remark 1 [10] By simple computation, it can easily be verified that

$$D_{n,q}(t^m;x)=\frac{{[n+1]}_q!}{{[n+m+1]}_q!}\sum _{k=0}^n{[k+1]}_q{[k+2]}_q\cdots {[k+m]}_q{p}_{nk}(q;x),r\ge 1.$$

Using ${[k+s]}_q={[s]}_q+q^s{[k]}_q$, we get ${[k+1]}_q{[k+2]}_q\cdots {[k+m]}_q={\prod }_{s=1}^m({[s]}_q+q^s{[k]}_q)={\sum }_{s=1}^m{c}_s(m){[k]}_q^s$, where $c_s(m)>0$, $s=0,1,2,\dots ,m$ are constants independent of k. Hence, $D_{n,q}(t^m;x)=\frac{{[n+1]}_q!}{{[n+m+1]}_q!}{\sum }_{s=1}^m{c}_s(m){\sum }_{k=0}^n{[k]}_q^s{p}_{nk}(q;x)=\frac{{[n+1]}_q!}{{[n+m+1]}_q!}{\sum }_{s=1}^m{c}_s(m){[n]}_q^s{B}_{n,q}(t^s;x)$.

Remark 2 For all $m\in \mathbf{N}\cup \{0\}$, $0\le \alpha \le \beta$, we have the following recursive relation for the images of the monomials $t^m$ under $D_{n,q}^{\alpha ,\beta }(t^m;x)$ in terms of $D_{n,q}(t^j;x)$; $j=0,1,2,\dots ,m$, as

$$\begin{array}{rl}{D}_{n,q}^{\alpha ,\beta }(t^m;x)& =\sum _{j=0}^m\left(\begin{array}{c}m\\ j\end{array}\right)\frac{{[n]}_q^j{\alpha }^{m-j}}{{({[n]}_q+\beta )}^m}{D}_{n,q}(t^j,x)\\ =\sum _{j=0}^m\left(\begin{array}{c}m\\ j\end{array}\right)\frac{{[n]}_q^j{\alpha }^{m-j}}{{({[n]}_q+\beta )}^m}\frac{{[n+1]}_q!}{{[n+j+1]}_q!}\sum _{s=1}^j{c}_s(m){[n]}_q^s{B}_{n,q}(t^s;x).\end{array}$$

Since $c_s(m)>0$ for $s=1,2,\dots ,m$ and $B_{n,q}(t^s;x)$ is a polynomial of degree $\le min(s,n)$ (see [13]), we get $D_{n,q}^{\alpha ,\beta }(t^m;x)$ is a polynomial of degree $\le min(m,n)$.

3 Convergence of q-Durrmeyer-Stancu operators

Theorem 1 Let $q_n\in (0,1]$. Then the sequence $\{D_{n,q_n}^{\alpha ,\beta }(f)\}$ convergence to f uniformly on $[0,1]$ for each $f\in C[0,1]$ if and only if ${lim}_{n\to \mathrm{\infty }}{q}_n=1$.

The proof of the above theorem follows along the lines of [10], Theorem 2], thus we omit the details.

Let $q\in (0,1)$ be fixed. We define $D_{\mathrm{\infty },q}^{\alpha ,\beta }(f,1)=f(1)$ and for $x\in (0,1)$

$$\begin{array}{rl}{D}_{\mathrm{\infty },q}^{\alpha ,\beta }(f;x)& =\frac{1}{1-q}\sum _{k=0}^{\mathrm{\infty }}{p}_{\mathrm{\infty }k}(q;x)q^{-k}{\int }_0^{1}f\left(\frac{{[n]}_{q}t+\alpha }{{[n]}_q+\beta }\right)p_{\mathrm{\infty }k}(q;qt)d_{q}t\\ =\sum _{k=0}^{\mathrm{\infty }}{A}_{\mathrm{\infty }k}^{\alpha ,\beta }(f)p_{\mathrm{\infty }k}(q;x).\end{array}$$

(4)

Using the fact that [14], we have

$$\begin{array}{c}\sum _{k=0}^{\mathrm{\infty }}{p}_{\mathrm{\infty }k}(q;x)=1,\sum _{k=0}^{\mathrm{\infty }}(1-q^k)p_{\mathrm{\infty }k}(q;x)=x,\text{and}\hfill \\ \sum _{k=0}^{\mathrm{\infty }}{(1-q^k)}^2{p}_{\mathrm{\infty }k}(q;x)=x^2+(1-q)x(1-x).\hfill \end{array}$$

Using (3) and (4), it is easy to prove that

$$\begin{array}{c}{D}_{\mathrm{\infty },q}^{\alpha ,\beta }(1;x)=1,D_{\mathrm{\infty },q}^{\alpha ,\beta }(t;x)=\frac{{[n]}_q(1+q(x-1))+\alpha }{{[n]}_q+\beta },\hfill \\ D_{\mathrm{\infty },q}^{\alpha ,\beta }(t^2;x)=\frac{{[n]}_q^2{q}^4{x}^2+({[n]}_q^2(q(1+q)(1-q^2))+2q\alpha {[n]}_q)x}{{({[n]}_q+\beta )}^2}\hfill \\ \phantom{D_{\mathrm{\infty },q}^{\alpha ,\beta }(t^2;x)=}+\frac{{[n]}_q^2{(1-q)}^2(1+q)+2\alpha (1-q){[n]}_q+{\alpha }^2}{{({[n]}_q+\beta )}^2}.\hfill \end{array}$$

For $f\in C[0,1]$, $t>0$, we define the modulus of continuity $\omega (f,t)$ as follows: $\omega (f,t)=sup\{|f(x)-f(y)|:|x-y|\le t,x,y\in [0,1]\}$. We shall show the following theorem.

Theorem 2 Let $0<q<1$ then for each $f\in C[0,1]$ the sequence $\{D_{n,q}^{\alpha ,\beta }(f;x)\}$ converges to $D_{\mathrm{\infty },q}^{\alpha ,\beta }(f;x)$ uniformly on $[0,1]$. Furthermore, $\parallel D_{n,q}^{\alpha ,\beta }(f)-D_{\mathrm{\infty },q}^{\alpha ,\beta }(f)\parallel \le C_q^{\alpha ,\beta }\omega (f,q^n)$.

The proof of the above theorem follows along the lines of [10], Theorem 3], thus we omit the details.

Remark 3 We may observe that, for $f(x)=x^2$, we have $\parallel D_{n,q}^{\alpha ,\beta }(f)-D_{\mathrm{\infty },q}^{\alpha ,\beta }(f)\parallel \asymp q^n\asymp {\omega }_2(f,\sqrt{q^n})$, where $A(n)\asymp B(n)$ means that $A(n)\ll B(n)$ and $A(n)\gg B(n)$, and $A(n)\ll B(n)$ means that there exists a positive constant C independent of n such that $A(n)\le CB(n)$. Hence, the estimate of Theorem 2 is sharp in the following sense: the sequence $q^n$ in Theorem 2 cannot be replaced by any other sequence decreasing to zero more rapidly as $n\to \mathrm{\infty }$.

Lemma 3 [15]

Let L be a positive linear operator on $C[0,1]$, which reproduces linear functions. If $L(t^2,x)>x^2$ $\mathrm{\forall }x\in (0,1)$, then $L(f)=f$ if and only if f is linear.

Remark 4 Since $D_{\mathrm{\infty },q}^{\alpha ,\beta }(t^2,x)=\frac{{[n]}_q^2[{(1-q)}^2(1+q)+q(1+2q)(1-q)x+q^3(1-q)x+q^4{x}^2]+2\alpha {[n]}_q(1+q(x-1))+{\alpha }^2}{{[n]}_q+\beta }>x^2$ for $0<q<1$ consequence of Lemma 3 we have the following:

Theorem 3 Let $0<q<1$ be fixed and let $f\in C[0,1]$. Then $D_{\mathrm{\infty },q}^{\alpha ,\beta }(f;x)=f(x)$ for all $x\in [0,1]$ if and only if f is linear.

Remark 5 Let $0<q<1$ be fixed and let $f\in C[0,1]$. Then the sequence $\{D_{n,q}^{\alpha \beta }(f;x)\}$ does not approximate $f(x)$ unless f is linear. This is completely in contrast to the classical Bernstein polynomials, by which $\{D_n(f;x)\}$ approximates $f(x)$ for any $f\in C[0,1]$.

Theorem 4 For any $f\in C[0,1]$, $\{D_{\mathrm{\infty },q}^{\alpha ,\beta }(f)\}$ converges to f uniformly on $[0,1]$ as $q\to 1-$.

Next, we establish a Voronovskaja type asymptotic formula for the operators $D_{n,q_n}^{\alpha ,\beta }$:

Theorem 5 Let f be bounded and integrable on the interval $[0,1]$, second derivative of f exists at a fixed point $x\in [0,1]$ and $q=q_n\in (0,1)$ such that $q_n\to 1$ as $n\to \mathrm{\infty }$, then

$$\underset{n\to \mathrm{\infty }}{lim}{[n]}_{q_n}(D_{n,q_n}^{\alpha ,\beta }(f;x)-f(x))=(\alpha +1-(2+\beta )x)f^{\prime }(x)+x(1+x)f^{″}(x).$$

The proof of the above lemma follows along the lines of [16], Theorem 3]; thus, we omit the details.