Genuine q-Stancu-Bernstein–Durrmeyer Operators

Abstract

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators $Z_n^{q,\alpha }(f;x)$. We calculate the moments of these operators, $Z_n^{q,\alpha }(t^j;x)$ for $j=0,1,2,$ which follows a symmetric pattern. We also calculate the second order central moment $Z_n^{q,\alpha }({(t-x)}^2;x).$ We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order $\beta$ and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order.

1. Introduction

A sequence of positive linear operators $S_n^{\left(\alpha \right)}:C[0,1]\to C[0,1]$ was introduced and studied by a Romanian mathematician D.D. Stancu [1] in 1968. The operators depend on a non-negative parameter $\alpha$ and are defined as follows:

$$S_n^{\left(\alpha \right)}(f,x)={\displaystyle \sum _{k=0}^n}f\left(\frac{k}{n}\right)p_{n,k}^{\left(\alpha \right)}\left(x\right),$$

where $p_{n,k}^{\left(\alpha \right)}\left(x\right)$ is the Polya distribution with density function given by

$$p_{n,k}^{\left(\alpha \right)}\left(x\right)=\left(\begin{array}{c}n\\ k\end{array}\right)\frac{{\displaystyle \prod _{i=0}^{k-1}}(x+i\alpha ){\displaystyle \prod _{s=0}^{n-k-1}}(1-x+s\alpha )}{{\displaystyle \prod _{i=0}^{n-1}}(1+i\alpha )},x\in [0,1].$$

If $\alpha =0,$ the operators $S_n^{\left(\alpha \right)}$ reduce to the well-known classical Bernstein polynomials. A. Lupaş and L. Lupaş [2] studied on a particular case for Stancu’s operators by setting $\alpha =\frac{1}{n}$ and obtained

$$S_n^{\left(\frac{1}{n}\right)}(f,x)=\frac{2(n!)}{(2n!)}{\displaystyle \sum _{k=0}^n}\left(\begin{array}{c}n\\ k\end{array}\right)f\left(\frac{k}{n}\right){\left(nx\right)}_k{(n-nx)}_{n-k,}$$

where ${\left(y\right)}_n=y(y+1)(y+2)\dots (y+n-1)$ is the rising factorial with ${\left(y\right)}_0=1.$

For any function $f\in C[0,1],$ $\alpha \ge 0$, $q>0$ and each $n\in \mathbb{N}$, modification of the Stancu’s operators based on the q-integers is introduced by G. Nowak [3] in 2009 as follows:

$$B_n^{q,\alpha }(f,x)={\displaystyle \sum _{k=0}^n}{p}_{n,k}^{q,\alpha }\left(x\right)f\left(\frac{{\left[k\right]}_q}{{\left[n\right]}_q}\right)x\in [0,1],$$

where

$$p_{n,k}^{q,\alpha }\left(x\right)={\left[\begin{array}{c}n\\ k\end{array}\right]}_q\frac{{\displaystyle \prod _{i=0}^{k-1}}(x+\alpha {\left[i\right]}_q){\displaystyle \prod _{s=0}^{n-k-1}}(1-q^{s}x+\alpha {\left[s\right]}_q)}{{\displaystyle \prod _{i=0}^{n-1}}(1+\alpha {\left[i\right]}_q)},$$

$$p_{n,k}^{q,\alpha }\left(1\right)=0,p_{n,n}^{q,\alpha }\left(1\right)=1,\prod _{s=0}^{-1}(\dots )=1.$$

G. Nowak studied the Korovkin type approximation properties for the q-analogue of the Stancu’s operators $B_n^{q,\alpha }(f,x)$. If $\alpha =0,$ $B_n^{q,\alpha }(f,x)$ reduces to the q-Bernstein polynomials defined by GM. Phillips in [4] as

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

where

$$p_{n,k}\left(q,x\right):={\left[\begin{array}{c}n\\ k\end{array}\right]}_q{x}^k\prod _{s=0}^{n-k-1}(1-q^{s}x).$$

For $q\to 1-,$ they reduce to Bernstein-Stancu operators. For $\alpha =0$ and $q\to 1-,$ they reduce to the classical Bernstein polynomials.

On the other hand, integral modification of classical Bernstein polynomials was introduced by J.L. Durrmeyer ([5]) in 1967. This type of modification was studied in several forms by different authors (see [6,7,8,9] and etc.). Classical genuine Bernstein-Durrmeyer operators were independently introduced by W. Chen ([10]) in 1987, and by T. N. T. Goodman & A. Sharma ([11]) later in 1991 and these operators were investigated by many authors, see for example [12,13]. They possess many interesting properties, in particular they reproduce linear functions and thus interpolate every function $f\in C[0,1]$ at $x=0$ and $x=1.$ V. Gupta ([14]) introduced the q-analogue of the Bernstein–Durrmeyer operators and studied some approximation properties for these operators.

To approximate continuous functions, V. Gupta and H. Wang ([15]) defined the q-Durrmeyer type operators as

$$M_{n,q}(f,x):=f\left(0\right)p_{n,0}(q,x)+{[n+1]}_q{\displaystyle \sum _{k=1}^n}{q}^{1-k}{p}_{n,k}\left(q;x\right){\int }_0^1{p}_{n,k-1}\left(q;qt\right)f\left(t\right)d_{q}t$$

and they studied estimation of the rate of convergence for continuous functions in terms of modulus of continuity. N. Mahmudov and P. Sabancıgil ([16]) defined and studied genuine q-Bernstein-Durrmeyer operators. T. Neer, P.N. Agrawal and S. Araci ([17]) constructed the Stancu-Durrmeyer type modification of q-Bernstein operators by means of q-Jackson integral. V. Gupta and Z. Finta ([18]) studied some direct local and global approximation theorems for the q-Durrmeyer operators for $0<q<1$. In [19,20] P. Sabancıgil shortly mentioned about some parts of the q-Stancu-Bernstein–Durrmeyer operators. Also, the rapid rise of post quantum calculus or shortly $(p,q)$-calculus has led to the discovery of new generalizations of this type of operators based on $(p,q)$-integers. M. Mursaleen et al. studied on $(p,q)$-analogue of Bernstein operators, some approximation results by $(p,q)$-analogue of Bernstein Stancu operators, approximation by $(p,q)$-Baskakov-Durrmeyer-Stancu operators, bivariate Bernstein-Schurer-Stancu type GBS operators in $(p,q)$-analogue and etc. (see [21,22,23,24]). Q.B. Chai and G. Zhou studied on $(p,q)$-analogue of Kantorovich type Bernstein-Stancu-Schurer operators (see [25]). V.N. Mishra studied Baskakov-Durrmeyer-Stancu operators (see [26]). I. Yuksel, U.D. Kantar and B. Altın studied on the $(p,q)$-Stancu generalization of a genuine Baskakov-Durrmeyer type operators (see [27]).

q-Stancu-Durrmeyer operators defined by T. Neer, P.N. Agrawal and S. Araci ([17]) do not preserve linear functions. The main motivation of this paper is to construct a q-analogue of Stancu-Bernstein–Durrmeyer type operators, $Z_n^{q,\alpha }(f;x),$ which preserves linear functions, i.e., $Z_n^{q,\alpha }(e_0;x)=e_0\left(x\right)$ and $Z_n^{q,\alpha }(e_1;x)=e_1\left(x\right)$ for $e_i\left(x\right)=x^i$ $(i=0,1)$ and interpolate every function $f\in C[0,1]$ at $x=0$ and $x=1.$

In this paper, we introduce genuine q-Stancu-Bernstein–Durrmeyer Operators $Z_n^{q,\alpha }$ in detail. We calculate the moments of these operators $Z_n^{q,\alpha }(t^j;x)$ for $j=0,1,2$ and observe that they follow a symmetrical pattern. We calculate the second order central moment $Z_n^{q,\alpha }({(t-x)}^2;x)$. We examine convergence properties of the operators, estimate the rate of convergence and we prove a Korovkin-type theorem. We also present a local approximation theorem by using the second order modulus of smoothness, we provide a local direct estimate in terms of Lipschitz type maximal function of order $\beta$ and we offer a global approximation theorem by using Ditzian-Totik modulus of second order. Compared to the previous generalizations of this type of operator, the advantage of genuine q-Stancu-Bernstein–Durrmeyer operators is that they reproduce linear functions and they interpolate every function $f\in C[0,1]$ at $x=0$ and $x=1$, i.e., $Z_n^{q,\alpha }(f;0)=f\left(0\right)$ and $Z_n^{q,\alpha }(f;1)=f\left(1\right).$

First of all, we provide some notations and definitions of q-integers:

Let $0<q<1.$ For any $n\in \mathbb{N}\cup \left\{0\right\}$, the q-integer ${\left[n\right]}_q$ is defined by

$${\left[n\right]}_q:=1+q+\dots +q^{n-1},{\left[0\right]}_q:=0;$$

and the q-factorial ${\left[n\right]}_q!$ by

$${\left[n\right]}_q!:={\left[1\right]}_q{\left[2\right]}_q\dots {\left[n\right]}_q,{\left[0\right]}_q!:=1.$$

For integers $0\le k\le n$, the q-binomial is defined by

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_q:=\frac{{\left[n\right]}_q!}{{\left[k\right]}_q!{\left[n-k\right]}_q!}.$$

The q-analogue of integration in the interval $[0,A]$ (see [28]) is defined by

$${\int }_0^{A}f\left(t\right)d_{q}t:=A\left(1-q\right)\sum _{n=0}^{\infty }f\left(A{q}^n\right)q^n,0<q<1.$$

For more information on q-calculus one can see [28].

The paper is organized as follows: In Section 2, we calculate the moments of these operators $Z_n^{q,\alpha }(t^j;x)$ for $j=0,1,2$ and the second order central moment $Z_n^{q,\alpha }({(t-x)}^2;x),$ we show that $Z_n^{q,\alpha }\left(t^m;x\right)$ is a polynomial of degree less than or equal to $min\left(m,n\right)$ and we prove that for $f\in C[0,1]$ and $0<q<1$, $Z_n^{q,\alpha }\left(f;x\right)=f\left(x\right)$ for all $x\in [0,1]$ if and only if f is linear. In Section 3, we examine convergence properties of the operators. In Section 4, we give an estimation for the rate of convergence, we prove a local approximation theorem by using the second order modulus of smoothness, we obtain a local direct estimate in terms of Lipschitz type maximal function of order $\beta$ and we prove a global approximation theorem by using second order Ditzian-Totik modulus. In Section 5, we present conclusions and suggest further studies.

2. Operators and Estimation of Their Moments

Definition 1.

For $f\in C\left[0,1\right]$, $\alpha \ge 0$ and $0<q<1,$ we define the following genuine q-Stancu-Bernstein–Durrmeyer Operators:

$$Z_n^{q,\alpha }\left(f;x\right)=p_{n,0}^{q,\alpha }\left(x\right)f\left(0\right)+p_{n,n}^{q,\alpha }\left(x\right)f\left(1\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\displaystyle \underset{0}{\overset{1}{\int }}}{p}_{n-2,k-1}(q;qt)f\left(t\right)d_{q}t.,$$

where for $n=1$ the sum is empty, i.e., equal to $0,$

$$p_{n,k}^{q,\alpha }\left(x\right)={\left[\begin{array}{c}n\\ k\end{array}\right]}_q\frac{{\displaystyle \prod _{i=0}^{k-1}}(x+\alpha {\left[i\right]}_q){\displaystyle \prod _{s=0}^{n-k-1}}(1-q^{s}x+\alpha {\left[s\right]}_q)}{{\displaystyle \prod _{i=0}^{n-1}}(1+\alpha {\left[i\right]}_q)},$$

$$p_{n,k}^{q,\alpha }\left(1\right)=0,p_{n,n}^{q,\alpha }\left(1\right)=1,\prod _{s=0}^{-1}(\dots )=1,$$

and

$$p_{n,k}\left(q,x\right):={\left[\begin{array}{c}n\\ k\end{array}\right]}_q{x}^k\prod _{s=0}^{n-k-1}(1-q^{s}x).$$

Moments and central moments play an important role in approximation theory. In the following lemma, we obtain explicit formulas for $Z_n^{q,\alpha }\left(t^m;x\right)$ for $m=0,1,2$ and $Z_n^{q,\alpha }\left({\left(t-x\right)}^2;x\right).$

Lemma 1.

We have

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(1;x\right)& =1,Z_n^{q,\alpha }\left(t;x\right)=x,\hfill \\ \hfill Z_n^{q,\alpha }\left(t^2;x\right)& =\frac{1}{{[n+1]}_q}x+\frac{q{\left[n\right]}_q}{{[n+1]}_q}\frac{1}{1+\alpha }\left(x(x+\alpha )+\frac{x(1-x)}{{\left[n\right]}_q}\right),\hfill \\ \hfill Z_n^{q,\alpha }\left({\left(t-x\right)}^2;x\right)& =\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x).\hfill \end{array}$$

Proof. 

By using the definition of q-Beta function (see [28]), we have the following equalities for $r=0,1,\dots$

$$\begin{array}{cc}\hfill {\int }_0^1{t}^r{p}_{n-2,k-1}\left(q;qt\right)d_{q}t& ={\left[\begin{array}{c}n-2\\ k-1\end{array}\right]}_q{q}^{k-1}{\int }_0^1{t}^{k+r-1}\prod _{r=0}^{n-k-2}(1-q^{r+1}t)d_{q}t\hfill \\ \hfill & =\frac{q^{k-1}{\left[n-2\right]}_q!}{{\left[k-1\right]}_q!{\left[n-1-k\right]}_q!}\frac{{\left[k+r-1\right]}_q!{\left[n-1-k\right]}_q!}{{\left[n+r-1\right]}_q!}\hfill \\ \hfill & =\frac{q^{k-1}{\left[n-2\right]}_q!{\left[k+r-1\right]}_q!}{{\left[k-1\right]}_q!{\left[n+r-1\right]}_q!}.\hfill \end{array}$$

(1)

We are going to use the following identities for the proof of the theorem (see [3]):

$${\displaystyle \sum _{k=0}^n}{p}_{n,k}^{q,\alpha }\left(x\right)=1,{\displaystyle \sum _{k=0}^n}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q}{{\left[n\right]}_q}=x,{\displaystyle \sum _{k=0}^n}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q^2}{{\left[n\right]}_q^2}=\frac{1}{1+\alpha }\left(x(x+\alpha )+\frac{x(1-x)}{{\left[n\right]}_q}\right).$$

Using Definition 1 and equality (1) and by using the above identities, we have

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(1;x\right)& =p_{n,0}^{q,\alpha }\left(x\right)+p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\displaystyle \underset{0}{\overset{1}{\int }}}{p}_{n-2,k-1}(q;qt)d_{q}t\hfill \\ \hfill & =p_{n,0}^{q,\alpha }\left(x\right)+p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{q^{k-1}}{{\left[n-1\right]}_q}\hfill \\ \hfill & =p_{n,0}^{q,\alpha }\left(x\right)+p_{n,n}^{q,\alpha }\left(x\right)+\sum _{k=1}^{n-1}{p}_{n,k}^{q,\alpha }\left(x\right)\hfill \\ \hfill & =\sum _{k=0}^n{p}_{n,k}^{q,\alpha }\left(x\right)=1.\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(t;x\right)& =p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\displaystyle \underset{0}{\overset{1}{\int }}}t{p}_{n-2,k-1}(q;qt)d_{q}t\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{q^{k-1}{\left[k\right]}_q}{{[n-1]}_q{\left[n\right]}_q}\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+\sum _{k=1}^{n-1}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q}{{\left[n\right]}_q}\hfill \\ \hfill & =\sum _{k=0}^n{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q}{{\left[n\right]}_q}=x\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(t^2;x\right)& =p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q{\displaystyle \sum _{k=1}^{n-1}}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\int }_0^1{t}^2{p}_{n-2,k-1}\left(q;qt\right)d_{q}t\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q{\displaystyle \sum _{k=1}^{n-1}}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{q^{k-1}{\left[n-2\right]}_q!{\left[k+1\right]}_q!}{{\left[k-1\right]}_q!{\left[n+1\right]}_q!}\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+\frac{1}{{\left[n+1\right]}_q}{\displaystyle \sum _{k=1}^{n-1}}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q(q{\left[k\right]}_q+1)}{{\left[n\right]}_q}\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+\frac{1}{{\left[n+1\right]}_q}{\displaystyle \sum _{k=0}^{n-1}}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q}{{\left[n\right]}_q}+\frac{q{\left[n\right]}_q}{{\left[n+1\right]}_q}{\displaystyle \sum _{k=0}^{n-1}}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k\right]}_q^2}{{\left[n\right]}_q^2}\hfill \\ \hfill & =p_{n,n}^{q,\alpha }\left(x\right)+\frac{1}{{\left[n+1\right]}_q}\left(x-p_{n,n}^{q,\alpha }\left(x\right)\right)+\frac{q{\left[n\right]}_q}{{\left[n+1\right]}_q}\left(\frac{1}{1+\alpha }\left(x(x+\alpha )+\frac{x(1-x)}{{\left[n\right]}_q}\right)-p_{n,n}^{q,\alpha }\left(x\right)\right)\hfill \\ \hfill & =\frac{1}{{\left[n+1\right]}_q}x+\frac{q{\left[n\right]}_q}{{\left[n+1\right]}_q}\frac{1}{1+\alpha }\left(x(x+\alpha )+\frac{x(1-x)}{{\left[n\right]}_q}\right).\hfill \end{array}$$

Now, for the proof of $Z_n^{q,\alpha }\left({(t-x)}^2;x\right),$ we use $Z_n^{q,\alpha }\left(t^2;x\right)$ and the linearity of the operators as follows:

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left({(t-x)}^2;x\right)& =Z_n^{q,\alpha }\left((t^2-2xt+x^2);x\right)\hfill \\ \hfill & =Z_n^{q,\alpha }\left(t^2;x\right)-2x{Z}_n^{q,\alpha }\left(t;x\right)+x^2\hfill \\ \hfill & =Z_n^{q,\alpha }\left(t^2;x\right)-x^2\hfill \\ \hfill & =\frac{1}{{[n+1]}_q}x+\frac{q{\left[n\right]}_q}{{[n+1]}_q}\frac{1}{1+\alpha }\left(x(x+\alpha )+\frac{x(1-x)}{{\left[n\right]}_q}\right)-x^2\hfill \\ \hfill & =\left(1-\frac{q{\left[n\right]}_q}{{[n+1]}_q}\frac{1}{1+\alpha }\right)x(1-x)+\frac{q}{{[n+1]}_q}\frac{1}{1+\alpha }x(1-x)+x\left(\frac{1+\alpha +q{\left[n\right]}_q(\alpha +1)}{{[n+1]}_q(1+\alpha )}-1\right)\hfill \\ \hfill & =\left(1-\frac{q{\left[n\right]}_q}{{[n+1]}_q}\frac{1}{1+\alpha }\right)x(1-x)+\frac{q}{{[n+1]}_q}\frac{1}{1+\alpha }x(1-x)\hfill \\ \hfill & =\left(\frac{{[n+1]}_q+\alpha {[n+1]}_q-\left({[n+1]}_q-1\right)+q}{{[n+1]}_q}\right)\frac{1}{1+\alpha }x(1-x)\hfill \\ \hfill & =\left(\frac{\alpha {[n+1]}_q+1+q}{{[n+1]}_q}\right)\frac{1}{1+\alpha }x(1-x)\hfill \\ \hfill & =\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x).\hfill \end{array}$$

Lemma is proved. □

Lemma 2.

$Z_n^{q,\alpha }\left(t^m;x\right)$ is a polynomial of degree less than or equal to $min\left(m,n\right)$.

Proof. 

By using simple calculations, we obtain

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(t^m;x\right)& ={\left[n-1\right]}_q{\displaystyle \sum _{k=1}^{n-1}}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\int }_0^1{p}_{n-2,k-1}\left(q;qt\right)t^m{d}_{q}t+p_{n,n}^{q,\alpha }\left(x\right)\hfill \\ \hfill & ={\left[n-1\right]}_q{\displaystyle \sum _{k=1}^{n-1}}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[n-2\right]}_q!{\left[k+m-1\right]}_q!}{{\left[k-1\right]}_q!{\left[n+m-1\right]}_q!}+p_{n,n}^{q,\alpha }\left(x\right)\hfill \\ \hfill & =\frac{{\left[n-1\right]}_q!}{{\left[n+m-1\right]}_q!}{\displaystyle \sum _{k=1}^{n-1}}{p}_{n,k}^{q,\alpha }\left(x\right)\frac{{\left[k+m-1\right]}_q!}{{\left[k-1\right]}_q!}+p_{n,n}^{q,\alpha }\left(x\right)\hfill \\ \hfill & =\frac{{\left[n-1\right]}_q!}{{\left[n+m-1\right]}_q!}{\displaystyle \sum _{k=1}^{n-1}}{\left[k\right]}_q{\left[k+1\right]}_q\dots {\left[k+m-1\right]}_q{p}_{n,k}^{q,\alpha }\left(x\right)+p_{n,n}^{q,\alpha }\left(x\right)\hfill \\ \hfill & =\frac{{\left[n-1\right]}_q!}{{\left[n+m-1\right]}_q!}{\displaystyle \sum _{k=1}^n}{\left[k\right]}_q{\left[k+1\right]}_q\dots {\left[k+m-1\right]}_q{p}_{n,k}^{q,\alpha }\left(x\right).\hfill \end{array}$$

Now using the identity that

$${\left[k\right]}_q{\left[k+1\right]}_q\dots {\left[k+m-1\right]}_q={\displaystyle \prod _{j=0}^{m-1}}\left(q^j{\left[k\right]}_q+{\left[j\right]}_q\right)=\sum _{j=1}^m{c}_j\left(m\right){\left[k\right]}_q^j,$$

where $c_j\left(m\right)>0$, $j=1,2,\dots ,m$, are the constants independent of k, we get

$$\begin{array}{cc}\hfill Z_n^{q,\alpha }\left(t^m;x\right)& =\frac{{\left[n-1\right]}_q!}{{\left[n+m-1\right]}_q!}{\displaystyle \sum _{k=1}^n}\sum _{j=1}^m{c}_j\left(m\right){\left[k\right]}_q^j{p}_{n,k}^{\alpha }\left(q;x\right)\hfill \\ \hfill & =\frac{{\left[n-1\right]}_q!}{{\left[n+m-1\right]}_q!}\sum _{j=1}^m{c}_j\left(m\right){\left[n\right]}_q^j{B}_{n,q}^{\alpha }\left(t^j;x\right).\hfill \end{array}$$

From [3], we know that $B_{n,q}^{\alpha }\left(t^j;x\right)$ is a polynomial of degree less than or equal to $min\left(j,n\right)$ and $c_j\left(m\right)>0$, $j=1,2,\dots ,m$. Thus it follows that $Z_n^{q,\alpha }\left(t^m;x\right)$ is a polynomial of degree less than or equal to $min\left(m,n\right)$. □

Theorem 1.

Let $f\in C[0,1]$ and $0<q<1$. Then $Z_n^{q,\alpha }\left(f;x\right)=f\left(x\right)$ for all $x\in [0,1]$ if and only if f is linear.

Proof. 

From Theorem 9 of [29], we know that for a positive linear operator D on $C[0,1]$ which reproduces linear functions, if $D(t^2,x)>x^2$ for all $x\in (0,1),$ then $D\left(f\right)=f$ if and only if f is linear. Now, since $Z_n^{q,\alpha }$ is a positive linear operator on $C[0,1]$ which reproduces linear functions, it sufficies to show that $Z_n^{q,\alpha }\left(t^2;x\right)>x^2$ for all $x\in (0,1).$

Let $h\left(x\right)=Z_n^{q,\alpha }\left(t^2;x\right)-x^2.$ One can easily see that $h\left(0\right)=h\left(1\right)=0.$ On the other hand we have,

$$\begin{array}{cc}\hfill h^{″}\left(x\right)& =\frac{q{\left[n\right]}_q}{{\left[n+1\right]}_q}\frac{2}{1+\alpha }-\frac{q}{{\left[n+1\right]}_q}\frac{2}{1+\alpha }-2\hfill \\ \hfill & =\frac{2q^2{\left[n-1\right]}_q}{{\left[n+1\right]}_q}\frac{1}{1+\alpha }-2<0.\hfill \end{array}$$

Now since the function h is concave down on $(0,1)$ and $h\left(0\right)=h\left(1\right)=0$, we conclude that $h\left(x\right)>0$ for all $x\in (0,1)$ which implies that $Z_n^{q,\alpha }\left(t^2;x\right)>x^2$ for all $x\in (0,1).$ This completes the proof of the theorem. □

3. Convergence of Genuine q-Stancu-Bernstein-Durrmeyer Operators

Theorem 2.

Let $\left(q_n\right)$ and $\left({\alpha }_n\right)$ be two sequences such that $0<q_n\le 1$ and ${\alpha }_n\ge 0$. Then the sequence $\left\{Z_n^{q_n,{\alpha }_n}\left(f\right)\right\}$ converges to f uniformly on $\left[0,1\right]$ for each $f\in C\left[0,1\right]$ if and only if ${lim}_{n\to \infty }{q}_n=1$ and ${lim}_{n\to \infty }{\alpha }_n=0.$

Proof. 

From the definition of $\left\{Z_n^{q,\alpha }\left(f\right)\right\}$ and Lemma 1 it follows that the operators $Z_n^{q_n,{\alpha }_n}$ are positive linear operators on $C\left[0,1\right]$ and reproduce linear functions. The well-known Korovkin theorem implies that $Z_n^{q_n,{\alpha }_n}\left(f\right)$ converges to $f\left(x\right)$ uniformly on $[0,1]$ as $n\to \infty$ for any $f\in C\left[0,1\right]$ if and only if

$$Z_n^{q_n,{\alpha }_n}\left(t^2;x\right)\to x^2$$

(2)

uniformly on $[0,1]$ as $n\to \infty$. If $q_n\to 1$ (${\left[n\right]}_{q_n}\to \infty )$ and ${\alpha }_n\to 0$ then (2) follows from Lemma 1.

On the other hand, if we assume that for any $f\in C\left[0,1\right]$, $Z_n^{q_n,{\alpha }_n}\left(f\right)$ converges to $f\left(x\right)$ uniformly on $[0,1]$ as $n\to \infty$, then $Z_n^{q_n,{\alpha }_n}\left(t^2;x\right)\to x^2.$ Hence

$$\frac{1}{{[n+1]}_{q_n}}x+\frac{q_n{\left[n\right]}_{q_n}}{{[n+1]}_{q_n}}\frac{1}{1+{\alpha }_n}\left(x(x+{\alpha }_n)+\frac{x(1-x)}{{\left[n\right]}_{q_n}}\right)\to x^2.$$

Consequently

$$\frac{q_n{\left[n\right]}_{q_n}}{{[n+1]}_{q_n}}\frac{1}{1+{\alpha }_n}\left(1-\frac{1}{{\left[n\right]}_{q_n}}\right)x^2+\left(\frac{1}{{[n+1]}_{q_n}}+\frac{q_n{\left[n\right]}_{q_n}}{{[n+1]}_{q_n}}\frac{{\alpha }_n}{1+{\alpha }_n}+\frac{q_n}{{[n+1]}_{q_n}}\frac{1}{1+{\alpha }_n}\right)x$$

converges to $x^2.$ Therefore

$$\begin{array}{cc}\hfill \frac{q_n{\left[n\right]}_{q_n}}{{[n+1]}_{q_n}}\frac{1}{1+{\alpha }_n}\left(1-\frac{1}{{\left[n\right]}_{q_n}}\right)& \to 1\mathrm{and}\hfill \\ \hfill \frac{1}{{[n+1]}_{q_n}}+\frac{q_n{\left[n\right]}_{q_n}}{{[n+1]}_{q_n}}\frac{{\alpha }_n}{1+{\alpha }_n}+\frac{q_n}{{[n+1]}_{q_n}}\frac{1}{1+{\alpha }_n}& \to 0.\hfill \end{array}$$

Since ${\alpha }_n$ and ${\left[n\right]}_{q_n}$ are both nonnegative we get ${\alpha }_n\to 0$ and ${\left[n\right]}_{q_n}\to \infty$ ($q_n\to 1$), which completes the proof of the theorem. □

Lemma 3.

For $f\in C[0,1],$ we have $∥Z_n^{q,\alpha }f∥\le ∥f∥.$

Proof. 

From Definition 1 and from Lemma 1, we have

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }\left(f;x\right)\right|& \le \left|f\left(0\right)\right|p_{n,0}^{q,\alpha }\left(x\right)+\left|f\left(1\right)\right|p_{n,n}^{q,\alpha }\left(x\right)+{\left[n-1\right]}_q\sum _{k=1}^{n-1}{q}^{1-k}{p}_{n,k}^{q,\alpha }\left(x\right){\int }_0^1{p}_{n-2,k-1}\left(q;qt\right)\left|f\left(t\right)\right|d_{q}t\hfill \\ \hfill & \le ∥f∥Z_n^{q,\alpha }\left(1;x\right)=∥f∥.\hfill \end{array}$$

□

4. Approximation Properties of q-Stancu-Bernstein–Durrmeyer Operators

We consider the following K-functional:

$$K_2\left(f,{\delta }^2\right):=inf\left\{∥f-h∥+{\delta }^2∥h^{″}∥:h\in C^2\left[0,1\right]\right\},\delta \ge 0,$$

where the space $C^2\left[0,1\right]$ is defined as

$$C^2\left[0,1\right]:=\left\{h:h,h^{\prime },h^{″}\in C\left[0,1\right]\right\}.$$

Then, from the well known result in [30], there exists an absolute constant $C_1>0$ such that

$$K_2\left(f,{\delta }^2\right)\le C_1{\omega }_2\left(f,\delta \right)$$

(3)

where

$${\omega }_2\left(f,\delta \right):=\underset{0<y\le \delta }{sup}\underset{x\pm y\in \left[0,1\right]}{sup}\left|f\left(x-y\right)-2f\left(x\right)+f\left(x+y\right)\right|$$

is the second modulus of smoothness of $f\in C\left[0,1\right]$.

In the following theorem, we state our first main result for this section. We prove a local approximation theorem for the genuine q-Stancu-Bernstein–Durrmeyer Operators $Z_n^{q,\alpha }$.

Theorem 3.

There exists an absolute constant $L>0$ such that

$$\left|Z_n^{q,\alpha }\left(f;x\right)-f\left(x\right)\right|\le L{\omega }_2\left(f,\sqrt{\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x\left(1-x\right)}\right),$$

where $f\in C[0,1]$, $0<q<1$.

Proof. 

Using the Taylor formula

$$h\left(t\right)=h\left(x\right)+h^{\prime }\left(x\right)\left(t-x\right)+{\int }_x^t\left(t-s\right)h^{″}\left(s\right)ds,h\in C^2\left[0,1\right],$$

we obtain that

$$Z_n^{q,\alpha }\left(h;x\right)=h\left(x\right)+Z_n^{q,\alpha }\left({\int }_x^t\left(t-s\right)h^{″}\left(s\right)ds;x\right),h\in C^2[0,1].$$

It is obvious that $\left|{\int }_x^t\left(t-s\right)h^{″}\left(s\right)ds\right|\le ∥h^{″}∥{\left(t-x\right)}^2.$ Hence

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }\left(h;x\right)-h\left(x\right)\right|& \le Z_n^{q,\alpha }\left(\left|{\int }_x^t\left|t-s\right|.\left|h^{″}\left(s\right)\right|ds\right|;x\right)\hfill \\ \hfill & \le ∥h^{″}∥Z_n^{q,\alpha }\left({\left(t-x\right)}^2;x\right)=∥h^{″}∥\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x).\hfill \end{array}$$

Now for $f\in C\left[0,1\right]$ and $h\in C^2\left[0,1\right]$ we obtain

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }\left(f;x\right)-f\left(x\right)\right|& \le \left|Z_n^{q,\alpha }\left(f-h;x\right)\right|+\left|Z_n^{q,\alpha }\left(h;x\right)-h\left(x\right)\right|+\left|f\left(x\right)-h\left(x\right)\right|\hfill \\ \hfill & \le 2∥f-h∥+∥h^{″}∥\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x).\hfill \end{array}$$

Taking the infimum on the right hand side over all $h\in C^2\left[0,1\right]$, we obtain

$$\left|Z_n^{q,\alpha }\left(f;x\right)-f\left(x\right)\right|\le 2K_2\left(f,\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x\left(1-x\right)\right).$$

(4)

Now the desired inequality follows from (3) and (4). □

We know that a function $f\in C[0,1]$ belongs to the Lipschitz type space $Li{p}_M\left(\beta \right),$ ($M>0$ and $0<\beta \le 1$), provided that

$$\left|f\left(t\right)-f\left(x\right)\right|\le M{\left|t-x\right|}^{\beta },\mathrm{for}\mathrm{all}x,t\in [0,1].$$

(5)

Then we have the following result.

Theorem 4.

Let $x\in [0,1].$ Then, for all $f\in Li{p}_M\left(\beta \right)$, $n\in \mathbb{N},$ $\alpha \ge 0$ and $0<q<1,$ we have

$$\left|Z_n^{q,\alpha }(f;x)-f\left(x\right)\right|\le M{\left\{{\lambda }_{\alpha ,n}(x,q)\right\}}^{\frac{\beta }{2}},$$

where

$${\lambda }_{\alpha ,n}(x,q)=\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x),$$

M is a constant depending on β and $f.$

Proof. 

Let $f\in Li{p}_M\left(\beta \right),$ $\left(0<\beta \le 1\right).$ By (5), we may write

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }(f;x)-f\left(x\right)\right|& \le Z_n^{q,\alpha }\left(\left|f\left(t\right)-f\left(x\right)\right|;x\right)\hfill \\ \hfill & \le M{Z}_n^{q,\alpha }\left({\left|t-x\right|}^{\beta }\right).\hfill \end{array}$$

Now, applying the Hölder inequality with $p=\frac{2}{\beta }$ and $q=\frac{2}{2-\beta }$, we get

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }(f;x)-f\left(x\right)\right|& \le M\left\{{\left[Z_n^{q,\alpha }\left({\left|t-x\right|}^{\beta p};x\right)\right]}^{\frac{1}{p}}{\left[Z_n^{q,\alpha }\left(1^q;x\right)\right]}^{\frac{1}{q}}\right\}\hfill \\ \hfill & =M\left\{{\left[Z_n^{q,\alpha }\left({\left|t-x\right|}^2;x\right)\right]}^{\frac{\beta }{2}}\right\}\hfill \\ \hfill & =M\left\{{\left[\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right)x(1-x)\right]}^{\frac{\beta }{2}}\right\}\hfill \\ \hfill & =M\left\{{\left[{\lambda }_{\alpha ,n}(x,q)\right]}^{\frac{\beta }{2}}\right\}\hfill \end{array}$$

and the proof is completed. □

In the next theorem, we give the direct global approximation theorem for the genuine q-Stancu-Bernstein–Durrmeyer Operators $Z_n^{q,\alpha }$. Before stating the theorem we recall the weighted K-functional of second order for $f\in C[0,1]$, it is defined by

$$K_{2,\varphi }\left(f,{\delta }^2\right):=inf\left\{∥f-g∥+{\delta }^2∥{\varphi }^2{g}^{″}∥:g\in W^2\left(\varphi \right)\right\},\delta \ge 0,{\varphi }^2=x\left(1-x\right)$$

where

$$W^2\left(\varphi \right):=\left\{g\in C\left[0,1\right]:g^{\prime }\in A{C}_{loc}\left[0,1\right],{\varphi }^2{g}^{″}\in C\left[0,1\right]\right\},$$

and $g^{\prime }\in A{C}_{loc}\left[0,1\right]$ means that g is differentiable and $g^{\prime }$ is absolutely continuous in every closed interval $\left[a,b\right]\subset \left[0,1\right]$. Furthermore, the Ditzian-Totik modulus of second order is given by

$${\omega }_2^{\varphi }\left(f,\delta \right):=\underset{0<g\le \delta }{sup}\underset{x\pm g\varphi \left(x\right)\in [0,1]}{sup}\left|f\left(x-\varphi \left(x\right)g\right)-2f\left(x\right)+f\left(x+\varphi \left(x\right)g\right)\right|.$$

It is well known that K-functional $K_{2,\varphi }\left(f,{\delta }^2\right)$ and Ditzian-Totik modulus ${\omega }_2^{\varphi }\left(f,\delta \right)$ are equivalent, see [30].

Theorem 5.

Let $f\in C[0,1]$, $0<q<1.$ There exists an absolute constant $L>0$ such that

$$∥Z_n^{q,\alpha }\left(f\right)-f∥\le L{\omega }_2^{\varphi }\left(f,\sqrt{\frac{\alpha }{1+\alpha }+\frac{1}{{[n+1]}_q(1+\alpha )}}\right).$$

Proof. 

From the Taylor expansion, we can write

$$h\left(t\right)=h\left(x\right)+h^{\prime }\left(x\right)\left(t-x\right)+{\int }_x^t{\int }_x^p{h}^{″}\left(s\right)dsdp.$$

Now, applying $Z_n^{q,\alpha }$ to both sides of the last equation we get

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }\left(h;x\right)-h\left(x\right)\right|& \le Z_n^{q,\alpha }\left(\left|{\int }_x^t{\int }_x^p\left|h^{″}\left(s\right)\right|dsdp\right|;x\right)\hfill \\ \hfill & \le ∥{\varphi }^2{h}^{″}∥Z_n^{q,\alpha }\left(\left|{\int }_x^t{\int }_x^p\left[\frac{1}{{\varphi }^2\left(s\right)}\right]dsdp\right|;x\right)\hfill \\ \hfill & \le ∥{\varphi }^2{h}^{″}∥Z_n^{q,\alpha }\left(\left|{\int }_x^t\frac{\left|t-s\right|}{{\varphi }^2\left(s\right)}ds\right|;x\right).\hfill \end{array}$$

Let $s=t+p\left(x-t\right),$ $p\in \left[0,1\right].$ Using the concavity of ${\varphi }^2$ we have

$$\frac{\left|t-s\right|}{{\varphi }^2\left(s\right)}=\frac{p\left|x-t\right|}{{\varphi }^2\left(t+p\left(x-t\right)\right)}\le \frac{p\left|x-t\right|}{{\varphi }^2\left(t\right)+p\left({\varphi }^2\left(x\right)-{\varphi }^2\left(t\right)\right)}\le \frac{\left|x-t\right|}{{\varphi }^2\left(x\right)}.$$

Therefore

$$\left|{\int }_x^t\frac{\left|t-s\right|}{{\varphi }^2\left(s\right)}ds\right|\le \left|{\int }_x^t\frac{\left|x-t\right|}{{\varphi }^2\left(x\right)}ds\right|=\frac{{\left(t-x\right)}^2}{{\varphi }^2\left(x\right)}$$

and

$$\left|Z_n^{q,\alpha }\left(h;x\right)-h\left(x\right)\right|\le ∥{\varphi }^2{h}^{″}∥\frac{1}{{\varphi }^2\left(x\right)}{Z}_n^{q,\alpha }\left({\left(t-x\right)}^2;x\right).$$

Since $Z_n^{q,\alpha }$ is a bounded operator, we obtain for $f\in C\left[0,1\right]$ that

$$\begin{array}{cc}\hfill \left|Z_n^{q,\alpha }\left(f;x\right)-f\left(x\right)\right|& \le \left|Z_n^{q,\alpha }\left(f-h;x\right)\right|+\left|Z_n^{q,\alpha }\left(h;x\right)-h\left(x\right)\right|+\left|f\left(x\right)-h\left(x\right)\right|\hfill \\ \hfill & \le 2∥f-h∥+∥{\varphi }^2{h}^{″}∥\left(\frac{\alpha }{1+\alpha }+\frac{1+q}{{[n+1]}_q(1+\alpha )}\right).\hfill \end{array}$$

Now, if we take the infimum on the right hand side over all h with $h^{\prime }\in A{C}_{loc}[0,1]$ we obtain

$$\left|Z_n^{q,\alpha }\left(f;x\right)-f\left(x\right)\right|\le 2K_{2,\varphi }\left(f,\left(\frac{\alpha }{1+\alpha }+\frac{1}{{[n+1]}_q(1+\alpha )}\right)\right).$$

Finally, from the fact that $K_{2,\varphi }(f,{\delta }^2)$ and ${\omega }_{\varphi }^2\left(f,\delta \right)$ are equivalent we obtain the desired result. □

5. Conclusions

Inspired from the previous studies, our goal was to construct a generalization of Stancu-Bernstein–Durrmeyer type operators based on the q-integers which reproduces linear functions and interpolates every function $f\in C[0,1]$ at $x=0$ and $x=1.$

In this paper, by using the q-analogue of integers, we defined the genuine q-Stancu-Bernstein–Durrmeyer Operators $Z_n^{q,\alpha }\left(f;x\right)$. We gave explicit formulas for the moments of these operators $Z_n^{q,\alpha }(t^j;x)$ for $j=0,1,2$ and for the second order central moment $Z_n^{q,\alpha }({(t-x)}^2;x)$. We proved a Korovkin-type theorem and we provided an estimation of the rate of convergence for these operators. We proved a local approximation theorem by using the second order modulus of smoothness, we obtained a local direct estimate in terms of Lipschitz-type maximal function of order $\beta$ and we proved a global approximation theorem by using Ditzian-Totik modulus of second order. The newly defined genuine q-Stancu-Bernstein–Durrmeyer operators have some advantages compared to the similar generalizations of this type of operator which was defined before. The advantage of genuine q-Stancu-Bernstein–Durrmeyer operators is that they preserve linear functions, i.e., $Z_n^{q,\alpha }(e_0;x)=e_0\left(x\right)$ and $Z_n^{q,\alpha }(e_1;x)=e_1\left(x\right)$ for $e_i\left(x\right)=x^i$ $(i=0,1)$ and furthermore, they interpolate every function $f\in C[0,1]$ at $x=0$ and $x=1$, i.e., $Z_n^{q,\alpha }(f;0)=f\left(0\right)$ and $Z_n^{q,\alpha }(f;1)=f\left(1\right).$ As a future work we would like to construct genuine $(p,q)$-Stancu-Bernstein–Durrmeyer operators $Z_n^{p,q,\alpha }\left(f;x\right)$ by using the theory of post quantum calculus and examine their approximation properties.