Approximation properties of bivariate extension of q-Szász-Mirakjan-Kantorovich operators

Abstract

In the present paper, a bivariate generalization of the q-Szász-Mirakjan-Kantorovich operators is constructed by $q_R$-integral and these operators’ weighted A-statistical approximation properties are established. Also, we estimate the rate of pointwise convergence of the proposed operators by modulus of continuity.

MSC:41A25, 41A36.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

In [1] the Kantorovich type generalization of the Szász-Mirakjan operators is defined by

$$K_n(f,x):=n{e}^{-nx}\sum _{k=0}^{\mathrm{\infty }}\frac{{(nx)}^k}{k!}{\int }_{k/n}^{(k+1)/n}f(t)dt,f\in C[0,\mathrm{\infty }),0\le x<\mathrm{\infty }.$$

In [2], for each positive integer n, Aral and Gupta defined q-type generalization of Szász-Mirakjan operators as

$$S_n^q(f;x):=E_q(-{[n]}_q\frac{x}{b_n})\sum _{k=0}^{\mathrm{\infty }}f\left(\frac{{[k]}_q{b}_n}{{[n]}_q}\right)\frac{{({[n]}_{q}x)}^k}{{[k]}_q!{(b_n)}^k},$$

where $0<q<1$, $f\in C[0,\mathrm{\infty })$, $0\le x<\frac{b_n}{(1-q){[n]}_q}$, $b_n$ is a sequence of positive numbers such that ${lim}_{x\to \mathrm{\infty }}{b}_n=\mathrm{\infty }$.

Recently, q-type generalization of Szász-Mirakjan operators, which was different from that in [2], was introduced, and the convergence properties of these operators were studied by Mahmudov [3]. Weighted statistical approximation properties of the modified q-Szász-Mirakjan operators were obtained in [4]. Also, Durrmeyer and Kantorovich-type generalizations of the linear positive operators based on q-integers were studied by some authors. The Bernstein-Durrmeyer operators related to the q-Bernstein operators were studied by Derriennic [5]. Gupta [6] introduced and studied approximation properties of q-Durrmeyer operators. The generalizations of the q-Baskakov-Kantorovich operators were constructed and weighted statistical approximation properties of these operators were examined in [7] and [8]. The q-extensions of the Szász-Mirakjan, Szász-Mirakjan-Kantorovich, Szász-Schurer and Szász-Schurer-Kantorovich operators were given shortly in [8]. Generalized Szász Durrmeyer operators were studied in [9]. With the help of $q_R$-integral, Örkcü and Doğru [10] introduced a Kantorovich-type modification of the q-Szász-Mirakjan operators as follows:

$$K_n^q(f;x):={[n]}_q{E}_q(-{[n]}_q\frac{x}{q})\sum _{k=0}^{\mathrm{\infty }}\frac{{({[n]}_{q}x)}^k}{{[k]}_q!q^k}{\int }_{q{[k]}_q/{[n]}_q}^{{[k+1]}_q/{[n]}_q}f(t)d_q^{R}t,$$

(1.1)

where $q\in (0,1)$, $0\le x<\frac{q}{1-q^n}$, $f\in C[0,\mathrm{\infty })$.

The paper of Mursaleen et al. [11] is one of the latest references on approximation by q-analogue. They investigated approximation properties for new q-Lagrange polynomials. Also, the q-analogue of Bernstein-Schurer-Stancu operators were introduced in [12].

On the other hand, Stancu [13] first introduced linear positive operators in two and several dimensional variables. Afterward, Barbosu [14] introduced a generalization of two-dimensional Bernstein operators based on q-integers and called them bivariate q-Bernstein operators. In recent years, many results have been obtained in the Korovkin-type approximation theory via A-statistical convergence for functions of several variables (for instance, [15–17]).

In this study, we construct a bivariate generalization of the Szász-Mirakjan-Kantorovich operators based on q-integers and obtain the weighted A-statistical approximation properties of these operators.

Now we recall some definitions about q-integers. For any non-negative integer r, the q-integer of the number r is defined by

$${[r]}_q=\{\begin{array}{cc}1+q+\cdots +q^{r-1}\hfill & \text{if }q\ne 1,\hfill \\ r\hfill & \text{if }q=1,\hfill \end{array}$$

where q is a positive real number. The q-factorial is defined as

$${[r]}_q!=\{\begin{array}{cc}{[1]}_q{[2]}_q\cdots {[r]}_q\hfill & \text{if }r=1,2,\dots ,\hfill \\ 1\hfill & \text{if }r=0.\hfill \end{array}$$

Two q-analogues of the exponential function $e^x$ are given as

$$\begin{array}{c}{E}_q(x)=\sum _{n=0}^{\mathrm{\infty }}{q}^{n(n-1)/2}\frac{x^n}{{[n]}_q!},x\in \mathbb{R},\hfill \\ {\epsilon }_q(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{{[n]}_q!},|x|<\frac{1}{1-q}.\hfill \end{array}$$

The following relation between q-exponential functions $E_q(x)$ and ${\epsilon }_q(x)$ holds

$$E_q(x){\epsilon }_q(-x)=1,|x|<\frac{1}{1-q}.$$

(1.2)

In the fundamental books about q-calculus (see [18, 19]), the q-integral of the function f over the interval $[0,b]$ is defined by

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

If f is integrable over $[0,b]$, then

$$\underset{q\to 1^{-}}{lim}{\int }_0^{b}f(t)d_{q}t={\int }_0^{b}f(t)dt.$$

A generally accepted definition for q-integral over an interval $[a,b]$ is

$${\int }_a^{b}f(t)d_{q}t={\int }_0^{b}f(t)d_{q}t-{\int }_0^{a}f(t)d_{q}t.$$

In order to generalize and spread the existing inequalities, Marinkovic et al. considered a new type of q-integral. So, the problems that ensue from the general definition of q-integral were overcome. The Riemann-type q-integral [20] in the interval $[a,b]$ was defined as

$${\int }_a^{b}f(t)d_q^{R}t=(1-q)(b-a)\sum _{j=0}^{\mathrm{\infty }}f(a+(b-a)q^j)q^j,0<q<1.$$

This definition includes only a point inside the interval of the integration.

2 Construction of the bivariate operators

The aim of this part is to construct a bivariate extension of the operators defined by (1.1).

For $n\in \mathbb{N}$, $0<q_1,q_2<1$ and $0\le x<\frac{q_{1_{}}}{1-q_1^n}$, $0\le y<\frac{q_{2_{}}}{1-q_2^n}$, the bivariate extension of the operators $K_n^q(f;x)$ is as follows:

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(f;x,y)& =& {[n]}_{q_1}{[n]}_{q_2}{E}_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{{({[n]}_{q_1}x)}^k}{{[k]}_{q_1}!q_1^k}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ \times {\int }_{q_2{[l]}_{q_2}/{[n]}_{q_2}}^{{[l+1]}_{q_2}/{[n]}_{q_2}}{\int }_{q_1{[k]}_{q_1}/{[n]}_{q_1}}^{{[k+1]}_{q_1}/{[n]}_{q_1}}f(t,s)d_{q_1}^{R}t{d}_{q_2}^{R}s,\end{array}$$

(2.1)

where

$$\begin{array}{rcl}{\int }_c^d{\int }_a^{b}f(t,s)d_{q_1}^{R}t{d}_{q_2}^{R}s& =& (1-q_1)(1-q_2)(b-a)(c-d)\\ \times \sum _{j=0}^{\mathrm{\infty }}\sum _{i=0}^{\mathrm{\infty }}f(a+(b-a)q_1^i,c+(c-d)q_2^j)q_1^i{q}_2^j.\end{array}$$

(2.2)

Also, f is a $q_R$-integrable function, so the series in (2.2) converges. It is clear that the operators given in (2.1) are linear and positive.

First, let us give the following lemma.

Lemma 1 Let $e_{ij}=x^i{y}^j$, $(i,j)\in {\mathbb{N}}_0\times {\mathbb{N}}_0$ with $i+j\le 2$ be the two-dimensional test functions. Then the following results hold for the operators given by (2.1):

$$\begin{array}{rc}\text{(i)}& K_n^{q_1,q_2}(e_{00};x,y)=1;\\ \text{(ii)}& K_n^{q_1,q_2}(e_{10};x,y)=x+\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}};\\ \text{(iii)}& K_n^{q_1,q_2}(e_{01};x,y)=y+\frac{1}{{[2]}_{q_2}}\frac{1}{{[n]}_{q_2}};\\ \text{(iv)}& K_n^{q_1,q_2}(e_{20};x,y)=q_1{x}^2+(q_1+\frac{2}{{[2]}_{q_1}})\frac{1}{{[n]}_{q_1}}x+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2};\\ \text{(v)}& K_n^{q_1,q_2}(e_{02};x,y)=q_2{y}^2+(q_2+\frac{2}{{[2]}_{q_2}})\frac{1}{{[n]}_{q_2}}y+\frac{1}{{[3]}_{q_2}}\frac{1}{{[n]}_{q_2}^2}.\end{array}$$

Proof From ${\int }_{q_2{[l]}_{q_2}/{[n]}_{q_2}}^{{[l+1]}_{q_2}/{[n]}_{q_2}}{\int }_{q_1{[k]}_{q_1}/{[n]}_{q_1}}^{{[k+1]}_{q_1}/{[n]}_{q_1}}{d}_{q_1}^{R}t{d}_{q_2}^{R}s=\frac{1}{{[n]}_{q_1}{[n]}_{q_2}}$ and the equality in (1.2), we have

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(e_{00};x,y)& =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{{({[n]}_{q_1}x)}^k}{{[k]}_{q_1}!q_1^k}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ =& 1.\end{array}$$

Since ${\int }_{q_2{[l]}_{q_2}/{[n]}_{q_2}}^{{[l+1]}_{q_2}/{[n]}_{q_2}}{\int }_{q_1{[k]}_{q_1}/{[n]}_{q_1}}^{{[k+1]}_{q_1}/{[n]}_{q_1}}t{d}_{q_1}^{R}t{d}_{q_2}^{R}s=\frac{1}{{[n]}_{q_1}{[n]}_{q_2}}(\frac{q_1{[k]}_{q_1}}{{[n]}_{q_1}}+\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}})$, we get from the linearity of $K_n^{q_1,q_2}$ that

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(e_{10};x,y)& =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{q_1{[k]}_{q_1}}{{[n]}_{q_1}}\frac{{({[n]}_{q_1}x)}^k}{{[k]}_{q_1}!q_1^k}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ +E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}\frac{{({[n]}_{q_1}x)}^k}{{[k]}_{q_1}!q_1^k}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}.\end{array}$$

Then we have from the definition of q-factorial and $K_n^{q_1,q_2}(e_{00};x,y)=1$

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(e_{10};x,y)& =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\frac{x{({[n]}_{q_1}x)}^{k-1}}{{[k-1]}_{q_1}!q_1^{k-1}}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}+\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}\\ =& x+\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}.\end{array}$$

Similarly, we write that

$$K_n^{q_1,q_2}(e_{01};x,y)=y+\frac{1}{{[2]}_{q_2}}\frac{1}{{[n]}_{q_2}}.$$

Now we compute the value $K_n^{q_1,q_2}(e_{20};x,y)$. Applying the equalities ${\int }_{q_2{[l]}_{q_2}/{[n]}_{q_2}}^{{[l+1]}_{q_2}/{[n]}_{q_2}}\times {\int }_{q_1{[k]}_{q_1}/{[n]}_{q_1}}^{{[k+1]}_{q_1}/{[n]}_{q_1}}{t}^2{d}_{q_1}^{R}t{d}_{q_2}^{R}s=\frac{1}{{[n]}_{q_1}{[n]}_{q_2}}(\frac{q_1^2{[k]}_{q_1}^2}{{[n]}_{q_1}^2}+\frac{1}{{[2]}_{q_1}}\frac{2q_1{[k]}_{q_1}}{{[n]}_{q_1}^2}+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2})$, $K_n^{q_1,q_2}(e_{10};x,y)=x+\frac{1}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}$ and $K_n^{q_1,q_2}(e_{00};x,y)=1$, we obtain

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(e_{20};x,y)& =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{q_1^2{[k]}_{q_1}^2}{{[n]}_{q_1}^2}\frac{{({[n]}_{q_1}x)}^k}{{[k]}_{q_1}!q_1^k}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ +\frac{2}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}x+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2}.\end{array}$$

Next, using the fact that ${[k]}_{q_1}=q_1{[k-1]}_{q_1}+1$, we obtain

$$\begin{array}{rcl}{K}_n^{q_1,q_2}(e_{20};x,y)& =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\frac{q_1{[k-1]}_{q_1}+1}{{[n]}_{q_1}^2}\frac{{({[n]}_{q_1}x)}^k}{{[k-1]}_{q_1}!q_1^{k-2}}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ +\frac{2}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}x+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2}\\ =& E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=2}^{\mathrm{\infty }}{q}_1{x}^2\frac{{({[n]}_{q_1}x)}^{k-2}}{{[k-2]}_{q_1}!q_1^{k-2}}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ +E_{q_1}(-{[n]}_{q_1}\frac{x}{q_1})E_{q_2}(-{[n]}_{q_2}\frac{y}{q_2})\\ \times \sum _{l=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\frac{q_1}{{[n]}_{q_1}}x\frac{{({[n]}_{q_1}x)}^{k-1}}{{[k-1]}_{q_1}!q_1^{k-1}}\frac{{({[n]}_{q_2}y)}^l}{{[l]}_{q_2}!q_2^l}\\ +\frac{2}{{[2]}_{q_1}}\frac{1}{{[n]}_{q_1}}x+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2}\\ =& q_1{x}^2+(q_1+\frac{2}{{[2]}_{q_1}})\frac{1}{{[n]}_{q_1}}x+\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2}.\end{array}$$

Similarly, we write

$$K_n^{q_1,q_2}(e_{02};x,y)=q_2{y}^2+(q_2+\frac{2}{{[2]}_{q_2}})\frac{1}{{[n]}_{q_2}}y+\frac{1}{{[3]}_{q_2}}\frac{1}{{[n]}_{q_2}^2},$$

which completes the proof. □

3 A-Statistical approximation properties

The Korovkin-type theorem for functions of two variables was proved by Volkov [21]. The theorem on weighted approximation for functions of several variables was proved by Gadjiev in [22].

Let $B_{\omega }$ be the space of real-valued functions defined on ${\mathbb{R}}^2$ and satisfying the bounded condition $|f(x,y)|\le M_f\omega (x,y)$, where $\omega (x,y)\ge 1$ for all $(x,y)\in {\mathbb{R}}^2$ is called a weight function if it is continuous on ${\mathbb{R}}^2$ and ${lim}_{\sqrt{x^2+y^2}\to \mathrm{\infty }}\omega (x,y)=\mathrm{\infty }$. We denote by $C_{\omega }$ the space of all continuous functions in the $B_{\omega }$ with the norm

$${\parallel f\parallel }_{\omega }=\underset{(x,y)\in {\mathbb{R}}^2}{sup}\frac{|f(x,y)|}{\omega (x,y)}.$$

Theorem 1 [22]

Let ${\omega }_1(x,y)$ and ${\omega }_2(x,y)$ be weight functions satisfying

$$\underset{\sqrt{x^2+y^2}\to \mathrm{\infty }}{lim}\frac{{\omega }_1(x,y)}{{\omega }_2(x,y)}=0.$$

Assume that $T_n$ is a sequence of linear positive operators acting from $C_{{\omega }_1}$ to $B_{{\omega }_2}$. Then, for all $f\in C_{{\omega }_1}$,

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel T_{n}f-f\parallel }_{{\omega }_2}=0$$

if and only if

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel T_n{F}_{\nu }-F_{\nu }\parallel }_{{\omega }_1}=0(\nu =0,1,2,3),$$

where $F_0(x,y)=\frac{{\omega }_1(x)}{1+x^2+y^2}$, $F_1(x,y)=\frac{x{\omega }_1(x)}{1+x^2+y^2}$, $F_2(x,y)=\frac{y{\omega }_1(x)}{1+x^2+y^2}$, $F_3(x,y)=\frac{(x^2+y^2){\omega }_1(x)}{1+x^2+y^2}$.

In [16], using the concept of A-statistical convergence, Erkuş and Duman investigated a Korovkin-type approximation result for a sequence of positive linear operators defined on the space of all continuous real-valued functions on any compact subset of the real m-dimensional space.

Now we recall the concepts of regularity of a summability matrix and A-statistical convergence. Let $A:=(a_{nk})$ be an infinite summability matrix. For a given sequence $x:=(x_k)$, the A-transform of x, denoted by $Ax:=({(Ax)}_n)$, is defined as ${(Ax)}_n:={\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}{x}_k$ provided the series converges for each n. A is said to be regular if ${lim}_n{(Ax)}_n=L$ whenever $limx=L$ [23]. Suppose that A is a non-negative regular summability matrix. Then x is A-statistically convergent to L if for every $\epsilon >0$, ${lim}_n{\sum }_{k:|x_k-L|\ge \epsilon }{a}_{nk}=0$, and we write ${\mathit{st}}_A\text{-}limx=L$ [24]. Actually, x is A-statistically convergent to L if and only if, for every $\epsilon >0$, ${\delta }_A(k\in \mathbb{N}:|x_k-L|\ge \epsilon )=0$, where ${\delta }_A(K)$ denotes the A-density of the subset K of the natural numbers and is given by ${\delta }_A(K):={lim}_n{\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}{\chi }_K(k)$ provided the limit exists, where ${\chi }_K$ is the characteristic function of K. If $A=C_1$, the Cesáro matrix of order one, then A-statistical convergence reduces to the statistical convergence [25]. Also, taking $A=I$, the identity matrix, A-statistical convergence coincides with the ordinary convergence.

We consider ${\omega }_1(x,y)=1+x^2+y^2$ and ${\omega }_2(x,y)={(1+x^2+y^2)}^{1+\alpha }$ for $\alpha >0$, $(x,y)\in {\mathbb{R}}_0^2$, where ${\mathbb{R}}_0^2:=\{(x,y)\in {\mathbb{R}}^2:x\ge 0,y\ge 0\}$.

We obtain statistical approximation properties of the operator defined by (2.1) with the help of Korovkin-type theorem given in [26]. Let $(q_{1,n})$ and $(q_{2,n})$ be two sequences in the interval $(0,1)$ so that

$$\begin{array}{r}{\mathit{st}}_A\text{-}\underset{n}{lim}{q}_{1,n}^n=1\text{and}{\mathit{st}}_A\text{-}\underset{n}{lim}{q}_{2,n}^n=1,\\ {\mathit{st}}_A\text{-}\underset{n}{lim}\frac{1}{{[n]}_{q_{1,n}}}=0\text{and}{\mathit{st}}_A\text{-}\underset{n}{lim}\frac{1}{{[n]}_{q_{2,n}}}=0.\end{array}$$

(3.1)

Theorem 2 Let $A=(a_{nk})$ be a nonnegative regular summability matrix, and let $(q_{1,n})$ and $(q_{2,n})$ be two sequences satisfying (3.1). Then, for any function $f\in C_{{\omega }_1}({\mathbb{R}}_0^2)$ and $q_R$-integrable function, for $\alpha >0$, we have

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\parallel K_n^{q_{1,n},q_{2,n}}f-f\parallel }_{{\omega }_2}=0.$$

Proof Let ${\tilde{K}}_n^{q_{1,n},q_{2,n}}$ be defined as

$${\tilde{K}}_n^{q_{1,n},q_{2,n}}(f;x,y)=\{\begin{array}{cc}{K}_n^{q_{1,n},q_{2,n}}(f;x,y),\hfill & 0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n},\hfill \\ f(x,y),\hfill & x\ge \frac{q_{1,n}}{1-q_{1,n}^n},y\ge \frac{q_{{2,n}_{}}}{1-q_{2,n}^n}.\hfill \end{array}$$

From Lemma 1, since $|K_n^{q_{1,n},q_{2,n}}(1+t^2+s^2;x,y)|\le c{(1+x^2+y^2)}^{1+\alpha }$ for $x\in [0,\frac{q_{1,n}}{1-q_{1,n}^n})$ and $y\in [0,\frac{q_{{2,n}_{}}}{1-q_{2,n}^n})$, $\{{\tilde{K}}_n^{q_{1,n},q_{2,n}}(f;\cdot )\}$ is a sequence of linear positive operators acting from $C_{{\omega }_1}({\mathbb{R}}_0^2)$ to $B_{{\omega }_2}({\mathbb{R}}_0^2)$.

From (i) of Lemma 1, it is clear that

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{00};\cdot )-e_{00}\parallel }_{{\omega }_1}=0$$

holds. By (ii) of Lemma 1, we get

$$\begin{array}{c}\underset{0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}}{sup}\frac{|K_n^{q_{1,n},q_{2,n}}(e_{10};\cdot )-e_{10}|}{1+x^2+y^2}\hfill \\ =\underset{0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}}{sup}\frac{|x+\frac{1}{{[2]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}}-x|}{1+x^2+y^2}\hfill \\ =\underset{0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}}{sup}\frac{1}{1+x^2+y^2}\frac{1}{{[2]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}}\hfill \\ =\frac{1}{{[2]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}}.\hfill \end{array}$$

Since ${\mathit{st}}_A\text{-}{lim}_n\frac{1}{{[n]}_{q_{1,n}}}=0$, ${\mathit{st}}_A\text{-}{lim}_n{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{10};\cdot )-e_{10}\parallel }_{{\omega }_1}=0$. Similarly, since ${\mathit{st}}_A\text{-}{lim}_n\frac{1}{{[n]}_{q_{2,n}}}=0$, ${\mathit{st}}_A\text{-}{lim}_n{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{01};\cdot )-e_{01}\parallel }_{{\omega }_1}=0$. Also, we have from (iv) of Lemma 1

$$\begin{array}{c}\underset{0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}}{sup}\frac{|K_n^{q_{1,n},q_{2,n}}(e_{20};\cdot )-e_{20}|}{1+x^2+y^2}\hfill \\ =\underset{0\le x<\frac{q_{1,n}}{1-q_{1,n}^n},0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}}{sup}\frac{|q_{1,n}{x}^2+(q_{1,n}+\frac{2}{{[2]}_{q_{1,n}}})\frac{1}{{[n]}_{q_{1,n}}}x+\frac{1}{{[3]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}^2}-x^2|}{1+x^2+y^2}\hfill \\ \le (1-q_{1,n})+(\frac{q_{1,n}}{2}+\frac{1}{{[2]}_{q_{1,n}}})\frac{1}{{[n]}_{q_{1,n}}}+\frac{1}{{[3]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}^2}.\hfill \end{array}$$

So, we can write

$${\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{20};\cdot )-e_{20}\parallel }_{{\omega }_1}\le (1-q_{1,n})+(\frac{q_{1,n}}{2}+\frac{1}{{[2]}_{q_{1,n}}})\frac{1}{{[n]}_{q_{1,n}}}+\frac{1}{{[3]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}^2}.$$

(3.2)

Since ${\mathit{st}}_A\text{-}{lim}_n(1-q_{1,n})=0$, ${\mathit{st}}_A\text{-}{lim}_n(\frac{q_{1,n}}{2}+\frac{1}{{[2]}_{q_{1,n}}})\frac{1}{{[n]}_{q_{1,n}}}=0$ and ${\mathit{st}}_A\text{-}{lim}_n\frac{1}{{[3]}_{q_{1,n}}}\frac{1}{{[n]}_{q_{1,n}}^2}=0$, for each $\epsilon >0$, we define the following sets.

$$\begin{array}{c}D:=\{k:{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{20};\cdot )-e_2\parallel }_{{\omega }_1}\ge \epsilon \},D_1:=\{k:1-q_{1,k}\ge \frac{\epsilon }{3}\},\hfill \\ D_2:=\{k:(\frac{q_{1,k}}{2}+\frac{1}{{[2]}_{q_{1,k}}})\frac{1}{{[n]}_{q_{1,k}}}\ge \frac{\epsilon }{3}\},D_3:=\{k:\frac{1}{{[3]}_{q_{1,k}}}\frac{1}{{[n]}_{q_{1,k}}^2}\ge \frac{\epsilon }{3}\}.\hfill \end{array}$$

By (3.2), it is clear that $D\subseteq D_1\cup D_2\cup D_3$, which implies that for all $n\in \mathbb{N}$,

$$\sum _{k\in D}{a}_{nk}\le \sum _{k\in D_1}{a}_{nk}+\sum _{k\in D_2}{a}_{nk}+\sum _{k\in D_3}{a}_{nk}.$$

Taking limit as $n\to \mathrm{\infty }$, we have

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{20};\cdot )-e_{20}\parallel }_{{\omega }_1}=0.$$

Similarly, since ${\mathit{st}}_A\text{-}{lim}_n(1-q_{2,n})=0$, ${\mathit{st}}_A\text{-}{lim}_n(\frac{q_{2,n}}{2}+\frac{1}{{[2]}_{q_{2,n}}})\frac{1}{{[n]}_{q_{2,n}}}=0$ and ${\mathit{st}}_A\text{-}{lim}_n\frac{1}{{[3]}_{q_{2,n}}}\times \frac{1}{{[n]}_{q_{2,n}}^2}=0$, we write ${\mathit{st}}_A\text{-}{lim}_n{\parallel {\tilde{K}}_n^{q_{1,n},q_{2,n}}(e_{02};\cdot )-e_{02}\parallel }_{{\omega }_1}=0$. So, the proof is completed. □

If we define the function ${\phi }_{x,y}(t,s)={(t-x)}^2+{(s-y)}^2$, $(x,y)\in [0,\frac{q_{1_{}}}{1-q_1^n})\times [0,\frac{q_{2_{}}}{1-q_2^n})$, then by Lemma 1 one gets the following result

$$\begin{array}{rcl}{K}_n^{q_1,q_2}({\phi }_{x,y}(t,s);x,y)& =& (q_1-1)x^2+(q_2-1)y^2+\frac{q_1}{{[n]}_{q_1}}x+\frac{q_2}{{[n]}_{q_2}}y\\ +\frac{1}{{[3]}_{q_1}}\frac{1}{{[n]}_{q_1}^2}+\frac{1}{{[3]}_{q_2}}\frac{1}{{[n]}_{q_2}^2}.\end{array}$$

We use the modulus of continuity $\omega (f,\delta )$ defined as follows:

$$\omega (f,\delta ):=sup\left\{\right|f(t,s)-f(x,y)|:(t,s),(x,y)\in {\mathbb{R}}_0^2\text{ and }\sqrt{{(t-x)}^2+{(s-y)}^2}\le \delta \},$$

where $\delta >0$ and $f\in C_B({\mathbb{R}}_0^2)$ the space of all bounded and continuous functions on ${\mathbb{R}}_0^2$. Observe that, for all $f\in C_B({\mathbb{R}}_0^2)$ and $\lambda ,\delta >0$, we have

$$\omega (f,\lambda \delta )\le (1+[\lambda ])\omega (f,\delta ),$$

(3.3)

where $[\lambda ]$ is defined to be the greatest integer less than or equal to λ.

By the definition of modulus of continuity, we have

$$|f(t,s)-f(x,y)|\le \omega (f,\sqrt{{(t-x)}^2+{(s-y)}^2}),$$

and by (3.3), for any $\delta >0$,

$$|f(t,s)-f(x,y)|\le (1+\left[\frac{\sqrt{{(t-x)}^2+{(s-y)}^2}}{\delta }\right])\omega (f,\delta ),$$

which implies that

$$|f(t,s)-f(x,y)|\le (1+\frac{{(t-x)}^2+{(s-y)}^2}{{\delta }^2})\omega (f,\delta ).$$

(3.4)

Using the linearity and positivity of the operators $K_n^{q_1,q_2}$, we get from (3.4) and $K_n^{q_1,q_2}(e_{00};x,y)=1$ that, for any $n\in \mathbb{N}$,

$$\begin{array}{rcl}|K_n^{q_1,q_2}(f;x,y)-f(x,y)|& \le & K_n^{q_1,q_2}\left(\right|f(t,s)-f(x,y)|;x,y)+|f(x,y)||K_n^{q_1,q_2}(e_{00};x,y)-e_{00}|\\ \le & K_n^{q_1,q_2}((1+\frac{{(t-x)}^2+{(s-y)}^2}{{\delta }^2})\omega (f,\delta );x,y)\\ =& (1+\frac{1}{{\delta }^2}{K}_n^{q_1,q_2}({\phi }_{x,y}(t,s);x,y))\omega (f,\delta ).\end{array}$$

Now, if we replace $q_{1,n}$ and $q_{2,n}$ by sequences $(q_{1,n})$ and $(q_{2,n})$ to be two sequences satisfying (3.1), and we take $\delta :={\delta }_n(x,y)=\sqrt{K_n^{q_{1,n},q_{2,n}}({\phi }_{x,y}(t,s);x,y)}$, $0\le x<\frac{q_{{1,n}_{}}}{1-q_{1,n}^n}$, $0\le y<\frac{q_{{2,n}_{}}}{1-q_{2,n}^n}$, then we can write

$$|K_n^{q_1,q_2}(f;x,y)-f(x,y)|\le 2\omega (f,\delta ).$$