The approximation of bivariate functions by bivariate operators and GBS operators

Ovidiu T. Pop$^\ast $

June 6, 2010.

$^\ast $National College “Mihai Eminescu", 5 Mihai Eminescu Street, 440014 Satu Mare,
Romania, e-mail: ovidiutiberiu@yahoo.com

In this paper we demonstrate a general approximation theorems for the bivariate functions by bivariate operators and GBS (Generalized Boolean Sum) operators.

MSC. 41A10, 41A25, 41A35, 41A36, 41A63.

Keywords. Linear positive operators, bivariate operators, GBS operators, Voronovskaja-type theorem, approximation theorem.

1 Introduction

In the paper [5], [7] we proved a Voronovskaja-type theorem and approximation theorem for a class of bivariate operators defined by finite sum, respectively by infinite sum. In the papers [8], and [9] we studied the approximation of bivariate functions by GBS operators.

The aim of this paper is to demonstrate a general approximation theorem of bivariate functions by special bivariate operators.

Let $\mathbb N$ be the set of positive integers and $\mathbb N_0=\mathbb N\cup \{ 0\} $.

In this section we recall some notions which we will use in this paper.

We consider $I\subset \mathbb R$, $I$ an interval and we shall use the function sets: $B(I)=\{ f|f:I\to \mathbb R$, $f$ bounded on $I\} $, $C(I)=\{ f|f:I\to \mathbb R$, $f$ continuous on $I\} $ and $C_B(I)=B(I)\cap C(I)$. For any $x\in I$, let the functions $\psi _x:I\to \mathbb R$, $\psi _x(t)=t-x$, for any $t\in I$ and $e_0:I\to \mathbb R$, $e_0(x)=1$ for any $x\in I$.

If $I\subset \mathbb R$ is a given interval and $f\in B(I)$, then the first order modulus of smoothness of $f$ is the function $\omega (f;\cdot ):[0,\infty )\to \mathbb R$ defined for any $\delta \geq 0$ by

\begin{equation} \label{1.1} \omega (f;\delta )=\sup \left\{ |f(x')-f(x'')|: x',\; x'\in I,\; |x'-x''|\leq \delta \right\} \! . \end{equation}

Let $I_1, I_2, J_1, J_2\subset \mathbb R$ be intervals, $E(I_1\times I_2)$, $F(J_1\times J_2)$ which are subsets of the set of real functions defined on $I_1\times I_2$, respectively $J_1\times J_2$ and $L:E(I_1\times I_2)\to F(J_1\times J_2)$ be a linear positive operator. The operator $UL:E(I_1\times I_2)\to F((I_1\cap J_1)\times (I_2\cap J_2))$ defined for any function $f\in E(I_1\times I_2)$, any $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$ by

\begin{equation} \label{1.2} (UL f)(x,y)=L\left(f(x, \ast )+f(\cdot , y)-f(\cdot , \ast )\right)(x,y) \end{equation}

is called GBS operator (“Generalized Boolean Sum" operator) associated to the operator $L$, where “$\cdot $" and “$\ast $" stand for the first and second variable (see [1]).

If $f\in E(I_1\times I_2)$ and $(x,y)\in I_1\times I_2$, let the functions $f_x=f(x, \ast )$, $f^y=f(\cdot , y):I_1\times I_2\to \mathbb R$, $f_x(s,t)=f(x,t)$, $f^y(s,t)=f(s,y)$ for any $(s,t)\in I_1\times I_2$. Then, we can consider that $f_x$, $f^y$ are functions of real variable, $f_x:I_2\to \mathbb R$, $f_x(t)=f(x,t)$ for any $t\in I_2$ and $f^y:I_1\to \mathbb R$, $f^y(s)=f^y(s,y)$ for any $s\in I_1$.

Let $I_1, I_2\subset \mathbb R$ be given intervals and $f:I_1\times I_2\to \mathbb R$ be a bounded function. The function $\omega _{total}(f;\, \cdot \, ,\ast ):[0,\infty )\times [0,\infty )\to \mathbb R$, defined for any $(\delta _1, \delta _2)\in [0,\infty )\times [0,\infty )$ by

\begin{align} \label{1.3} \omega _{total}(f; \delta _1, \delta _2) = & \sup \big\{ |f(x,y)-f(x’,y’)|:(x,y), (x’,y’)\in I_1\times I_2,\\ & \quad \quad \ |x-x’|\leq \delta _1, |y-y’|\leq \delta _2\big\} \nonumber \end{align}

is called the first order modulus of smoothness of function $f$ or total modulus of continuity of function $f$ (see [11]).

The first order modulus of smoothness for bivariate functions has properties similar to the properties of the first modulus of smoothness for univariate functions.

If $(L_m)_{m\geq 1}$ is a sequence of operators, $L_m:E(I)\to F(J)$, $m\in \mathbb N$, for $m\in \mathbb N$ and $i\in \mathbb N_0$ define $T_{m,i}$ by

\begin{equation} \label{1.4} (T_{m,i}L_m)(x)=m^i\left(L_m \psi ^i_x\right)(x) \end{equation}

for any $x\in I\cap J$, where $E(I)$, $F(J)$ are subsets of the set of real functions defined on $I$, respectively $J$.

2 Preliminaries

In this section let $p_m=m$ for any $m\in \mathbb N$ or $p_m=\infty $ for any $m\in \mathbb N$ and similarly is defined $q_n$, $n\in \mathbb N$.

Let $I_1, I_2, J_1, J_2\subset \mathbb R$ be intervals with $I_1\cap J_1\neq \emptyset $ and $I_2\cap J_2\neq \emptyset $. For $m,n\in \mathbb N$ and $k\in \{ 0,1,\dots , p_m\} \cap \mathbb N_0$, $j\in \{ 0,1,\dots , q_n\} \cap \mathbb N_0$, we consider $\varphi _{m,k}:J_1\to \mathbb R$, $\varphi _{m,k}(x)\geq 0$ for any $x\in J_1$, $\psi _{n,j}:J_2\to \mathbb R$, $\psi _{n,j}(y)\geq 0$ for any $y\in J_2$ and the linear positive functionals $A_{m,k}:E_1(I_1)\to \mathbb R$, $B_{n,j}:E_2(I_2)\to \mathbb R$.

Definition 2.1

For $m,n\in \mathbb N$ define the sequences of operators $(L_m)_{m\geq 1}$ and $(K_n)_{n\geq 1}$ by

\begin{equation} \label{2.1} (L_m f)(x)=\sum ^{p_m}_{k=0}\varphi _{m,k}(x) A_{m,k}(f), \end{equation}
5

\begin{equation} \label{2.2} (K_n g)(y)=\sum ^{q_n}_{j=0}\psi _{n,j}(y)B_{n,j}(g) \end{equation}
6

for any $f\in E_1(I_1)$, $g\in E_2(I_2)$, $x\in J_1$ and $y\in J_2$, where $E_1(I_1)$, $E_2(I_2)$ are subsets of the set of real functions defined on $I_1$, respectively $I_2$.

Proposition 2.2

The operators $(L_m)_{m\geq 1}$ and $(K_n)_{n\geq 1}$ are linear positive on $E_1(I_1\cap J_1)$ and $E_2(I_2\cap J_2)$ respectively.

Proof â–¼

The proof follows immediately.

In the following let $s\in \mathbb N_0$, $s$ even. We suppose that the operators $(L_m)_{m\geq 1}$, $(K_n)_{n\geq 1}$ verify the conditions: there exist the smallest $\alpha _j, \beta _j\in [0,\infty )$, $j\in \{ 0,2,4,\dots , s+2\} $, such that

\begin{equation} \label{2.3} \lim _{m\to \infty }\tfrac {(T_{m,j}L_m)(x)}{m^{\alpha _j}}= a_j(x) \end{equation}

for any $x\in I_1\cap J_1$,

\begin{equation} \label{2.4} \lim _{n\to \infty }\tfrac {(T_{n,j}K_n)(y)}{n^{\beta _j}}= b_j(y) \end{equation}

for any $y\in I_2\cap J_2$ and if we note

\begin{equation} \label{2.5} \gamma _s=\max \left\{ \alpha _{s-2l+\beta _{2l}}: l\in \left\{ 0,1,\dots , \tfrac {s}{2}\right\} \right\} \! , \end{equation}

then

\begin{equation} \label{2.6} \left\{ \begin{array}{l} \alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2<0\\ \alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2<0\\ \alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4<0 \end{array}\right. \end{equation}

where $l\in \left\{ 0,1,2,\dots \tfrac {s}{2}\right\} $.

In the following we consider the set $E(I_1\times I_2)=\big\{ f|f:I_1\times I_2\to \mathbb R$, $f_x\in E_2(I_2)$ for any $x\in I_1$ and $f^y\in E_1(I_1)$ for any $y\in I_2\big\} $.

For $m,n\in \mathbb N$, let the linear positive functionals $A_{m,n,k,j}:E(I_1\times I_2)\to \mathbb R$ with the property

\begin{equation} \label{2.7} A_{m,n,k,j}\left((\cdot -x)^i(\ast -y)^l\right)= A_{m,k}\left((\cdot -x)^i\right)B_{n,j}\left((\ast -y)^l\right) \end{equation}

for any $k\in \{ 0,1,\dots , p_m\} \cap \mathbb N_0$, $j\in \{ 0,1,\dots , q_n\} \cap \mathbb N_0$, $i,l\in \{ 0,1,\dots , s\} $ and $x\in I_1$, $y\in I_2$.

Definition 2.3

Let $m,n\in \mathbb N$. The operator $L^\ast _{m,n}$ defined for any function $f\in E(I_1\times I_2)$ and any $(x,y)\in J_1\times J_2$ by

\begin{equation} \label{2.8} \left(L_{m,n}^\ast f\right)(x,y)= \sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x) \psi _{n,j}(y)A_{m,n,k,j}(f) \end{equation}
12

is named the bivariate operator of $LK$-type.

Proposition 2.4

The operators $\left(L_{m,n}^\ast \right)_{m,n\geq 1}$ are linear and positive on $E\left((I_1\cap J_1)\times (I_2\cap J_2)\right)$.

Proof â–¼

The proof follows immediately.

In the following we consider that

\begin{equation} \label{2.9} (T_{m,0}L_m)(x)=A_{m,0}(e_0)=1 \end{equation}

for any $x\in I_1\cap J_1$, $m\in \mathbb N$ and

\begin{equation} \label{2.10} (T_{n,0}K_n)(y)=B_{n,0}(e_0)=1 \end{equation}

for any $y\in I_2\cap J_2$, $n\in \mathbb N$.

From 13, 14 it results immediately that

\begin{equation} \label{2.11} \sum ^{p_m}_{k=0}\varphi _{m,k}(x)=1 \end{equation}

for any $x\in I_1\cap J_1$, $m\in \mathbb N$ and

\begin{equation} \label{2.12} \sum ^{q_n}_{j=0}\psi _{n,j}(y)=1 \end{equation}

for any $y\in I_2\cap J_2$, $n\in \mathbb N$.

Remark 2.5

From 13 and 14 it results that $\alpha _0=\beta _0=0$.â–¡

3 Main results

We recall the following theorem from [5].

Theorem 3.1

Let $I_1, I_2\subset \mathbb R$ be intervals, $(a,b)\in I_1\times I_2$, $n\in \mathbb N_0$ and the function $f:I_1\times I_2\to \mathbb R$, $f$ admits partial derivatives of order $n$ continuous in a neighborhood $V$ of the point $(a,b)$. According to Taylor’s expansion theorem for the function $f$ around $(a,b)$, for $(x,y)\in V$ we have

\begin{align} \label{3.1} f(x,y)& =\sum ^n_{k=0}\tfrac {1}{k!}\left(\tfrac {\partial }{\partial x}\, (x-a)+\tfrac {\partial }{\partial y}\, (y-b)\right)^k f(a,b)+\\ & \quad +\rho ^n(x,y)\mu (x-a, y-b)\nonumber \end{align}

where

\begin{align} \label{3.2} & \left(\tfrac {\partial }{\partial x}\, (x-a)+\tfrac {\partial }{\partial y}\, (y-b)\right)^kf(a,b)=\\ & =\sum ^k_{i=0}\tbinom {k}{i}\tfrac {\partial ^k f}{\partial x^{k-i} \partial y^i}\, (a,b)(x-a)^{k-i}(y-b)^i,\nonumber \end{align}

$k\in \{ 0,1,\dots ,n\} $, $\mu $ is a bounded function with $\lim \limits _{(x,y)\to (a,b)}\mu (x-a, y-b)=0$ and

\begin{equation} \label{3.3} \rho (x,y)=\sqrt{(x-a)^2+(y-b)^2}. \end{equation}

Then for any $\delta _1, \delta _2{\gt}0$, any $(x,y)\in V$ we have

\begin{multline} \label{3.4} \left|\mu (x-a, y-b)\right|\leq \\ \qquad \leq \tfrac {1}{n!}\left(1\! +\! \delta ^{-2}_1(x\! -\! a)^2\right)\! \left(1\! +\! \delta _2^{-2}(y\! -\! b)^2\right)\sum ^n_{i=0}\tbinom {n}{i} \omega _{\text{total}}\left(\! \tfrac {\partial ^n f}{\partial x^{n-i}\partial y^i};\delta _1, \delta _2\! \right)\! . \end{multline}

Theorem 3.2

Let $f:I_1\times I_2\to \mathbb R$ be a bivariate function.
If $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$ and $f$ admits partial derivatives of order $s$ continuous in a neighborhood of the point $(x,y)$, then

\begin{align} \label{3.5} & \lim _{m\to \infty }m^{s-\gamma _s}\bigg[\left(L^\ast _{m,m}f\right) (x,y)-\\ & \quad \quad \quad \quad \quad \ -\sum ^s_{i=0}\tfrac {1}{m^i i!}\sum ^i_{l=0} \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}\, (x,y) \left(T_{m,i-l}L_m\right)(x) \left(T_{m,l}K_m\right)(y)\bigg]=0.\nonumber \end{align}

If $f$ admits partial derivatives of order $s$ continuous on $(I_1\cap J_1)\times (I_2\cap J_2)$ and there exist the intervals $K_1\subset I_1\cap J_1$, $K_2\subset I_2\cap J_2$ such that there exist $m(s)\in \mathbb N$ and $a_{2l}, b_{2l}\in \mathbb R$ depending on $K_1$, respectively $K_2$, so that for any $m\in \mathbb N$, $m\geq m(s)$ and for any $(x,y)\in K_1\times K_2$ we have

\begin{equation} \label{3.6} \tfrac {(T_{m, 2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}

\begin{equation} \label{3.7} \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l}, \end{equation}

where $l\in \left\{ 0,1,\dots , \tfrac {s}{2}+1\right\} $, then the convergence given in $(\ref{3.5})$ is uniform on $K_1\times K_2$ and

\begin{align} \label{3.8} & m^{s-\gamma _s}\bigg|\left(L^\ast _{m,m}f\right)(x,y)-\\ & \quad \quad \quad -\sum ^s_{i=0}\tfrac {1}{m^i i!}\sum ^i_{l=0} \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}\, (x,y) \left(T_{m,i-l}L_m\right)(x)\left(T_{m,l}K_m\right)(y) \bigg|\leq \nonumber \\ & \leq \tfrac {1}{s!}\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l} (a_{s-2l}+a_{s-2l+2})(b_{2l}+b_{2l+2})\cdot \nonumber \\ & \quad \cdot \sum ^s_{i=0}\tbinom {s}{i}\omega _{\text{total}} \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ; \tfrac {1}{\sqrt{m^{\delta _s}}}\, ,\tfrac {1}{\sqrt{m^{\delta _s}}}\right)\nonumber \end{align}

for any $(x,y)\in K_1\times K_2$, any $m\in \mathbb N$ with $m\geq m(s)$, where

\begin{align} \label{3.9} \delta _s = & -\max \bigg\{ \alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2, \alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2,\\ & \quad \quad \quad \quad \tfrac {1}{2}\, (\alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4): l\in \Big\{ 0,1,\dots ,\tfrac {s}{2} \Big\} \! \bigg\} .\nonumber \end{align}

Proof â–¼

Let $m,n\in \mathbb N$. According to Taylor’s formula for the function $f$ around $(x,y)$, we have

\[ f(t, \tau )=\sum ^s_{i=0}\tfrac {1}{i!}\left(\tfrac {\partial }{\partial t}(t-x)+\tfrac {\partial }{\partial \tau }(\tau -y)\right)^i f(x,y)+\rho ^s(t,\tau )\mu (t-x, \tau -y), \]

from where

\begin{align} \label{3.10} f(t, \tau )& =\sum ^s_{i=0}\tfrac {1}{i!}\sum ^i_{l=0} \tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y) (t-x)^{i-l}(\tau -y)^l+\\ & \quad +\rho ^s(t,\tau )\mu (t-x, \tau -y),\nonumber \end{align}

where $\mu $ is a bounded function and $\lim \limits _{(t,\tau )\to (x,y)}\mu (t-x, \tau -y)=0$. Because $A_{m,n,k,j}$ is linear positive functional and verifies (11), from (27) we have

\begin{align*} A_{m,n,k,j}(f) & =\sum ^s_{i=0}\tfrac {1}{i!}\sum ^i_{l=0}\tbinom {i}{l} \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}\, (x,y) A_{m,k}\left((\cdot \! -\! x)^{i-l}\right)B_{n,j}\left((\ast \! -\! y)^l\right)\! +\\ & \quad +A_{m,n,k,j}\left(\rho ^s(\cdot , \ast )\mu _{xy}\right), \end{align*}

where $\mu _{xy}:(I_1\cap J_1)\times (I_2\cap J_2)\to \mathbb R$, $\mu _{x,y}(t,\tau )=\mu (t-x, \tau -y)$ for any $(t, \tau )\in (I_1\cap J_1)\times (I_2\cap J_2)$. Multiplying by $\varphi _{m,k}(x)\psi _{n,j}(y)$ and summing after $k,j$ where $k\in \{ \! 0,1,\dots , p_m\! \} \cap \mathbb N_0$, $j\in \{ 0,1,\dots , q_n\} \cap \mathbb N_0$, we obtain

\begin{align*} (L^\ast _{m,n}f)(x,y) & =\sum ^s_{i=0}\tfrac {1}{i!}\sum ^i_{l=0}\tbinom {i}{l} \tfrac {\partial ^i f}{\partial ^{i-l}\partial \tau ^l}\, (x,y) \tfrac {1}{m^{i-l}}\, \tfrac {1}{n^l}(T_{m,i-l}L_m)(x)\cdot \\ & \quad \cdot (T_{n,l}K_n)+\sum ^{p_m}_{k=0}\sum ^{q_m}_{j=0} \varphi _{m,k}(x)\psi _{n,j}(y)A_{m,n,k,j}\left(\rho ^s(\cdot , \ast )\mu _{xy}\right), \end{align*}

from which

\begin{align} \label{3.11} m^{s-\gamma _s} & \bigg[(L^\ast _{m,m}f)(x,y)-\\ & \quad -\sum ^s_{i=0}\tfrac {1}{m^i i!}\sum ^i_{l=0}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y) (T_{m,i-l}L_m)(x)(T_{m,l}K_m)(y)\bigg]=\nonumber \\ & =(R_{m,m}f)(x,y),\nonumber \end{align}

where

\begin{equation} \label{3.12} (R_{m,m}f)(x,y)\! =\! m^{s-\gamma _s}\sum ^{p_m}_{k=0}\sum ^{q_m}_{j=0} \varphi _{m,k}(x)\psi _{m,j}(y)A_{m,m,k,j}\left(\rho ^s(\cdot , \ast )\mu _{xy}\right). \end{equation}

Then

\[ |(R_{m,m}f)(x,y)|\leq m^{s-\gamma _s}\sum ^{p_m}_{k=0} \sum ^{q_m}_{j=0}\varphi _{m,k}(x)\psi _{m,j}(y)\big|A_{m,m,k,j}\left(\rho ^s(\cdot , \ast )\mu _{xy}\right)\big|, \]

from where

\begin{align} \label{3.13} & |(R_{m,m}f)(x,y)|\leq \\ & \leq m^{s-\gamma _s} \sum ^{p_m}_{k=0}\sum ^{q_m}_{j=0}\varphi _{m,k}(x) \psi _{m,j}(y)A_{m,m,k,j}\left(\rho ^s(\cdot , \ast )|\mu _{xy}|\right)\! .\nonumber \end{align}

According to the relation (20), for any $\delta _1, \delta _2{\gt}0$ and for any $(t, \tau )\in (I_1\cap J_1)\times (I_2\cap J_2)$, we have that

\begin{align*} |\mu _{xy}(t,\tau )|& =|\mu (t-x, \tau -y)|\leq \tfrac {1}{s!}\big(1+\delta _1^{-2}(t-x)^2+ \delta _2^{-2}(\tau -y)^2+\\ & \quad +\delta _1^{-2}\delta _2^{-2}(t-x)^2(\tau -y)^2\big)\sum ^s_{i=0}\tbinom {s}{i}\omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ,\delta _1, \delta _2\right) \end{align*}

and taking $\rho ^s(t, \tau )=\displaystyle \sum ^{\tfrac {s}{2}}_{l=0}\displaystyle \tbinom {\tfrac {s}{2}}{l} (t-x)^{s-2l}(\tau -y)^{2l}$ into account, (31) results

\begin{multline} \label{3.14} A_{m,m,k,j}\left(\rho ^s(\cdot , \ast )|\mu _{xy}|\right)\leq \tfrac {1}{s!}\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l}\Big[A_{m,k}\big(\psi ^{s-2l}_x\big) B_{m,j}\big(\psi ^{2l}_y\big)+\\ +\delta _1^{-2}A_{m,k}\big(\psi ^{s-2l+2}_x\big)B_{m,j}\big(\psi ^{2l}_y\big)+ \delta _2^{-2}A_{m,k}\big(\psi ^{s-2l}_x\big)B_{m,j}\big(\psi ^{2l+2}_y\big)+\\ \qquad \quad +\delta _1^{-2}\delta _2^{-2}A_{m,k}\big(\psi ^{s-2l+2}_x\big)B_{m,j} \big(\psi ^{2l+2}_y\big)\Big] \sum ^s_{i=0}\tbinom {s}{i}\omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}; \delta _1, \delta _2\! \right)\! . \end{multline}

From (30) and (31), it results that

\begin{align*} & |(R_{m,m}f)(x,y)|\leq \\ & \leq \tfrac {1}{s!}m^{s-\gamma _s}\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l} \sum ^{p_m}_{k=0}\sum ^{q_m}_{j=0}\varphi _{m,k}(x)\psi _{m,j}(y) \Big[A_{m,k}\big(\psi ^{s-2l}_x\big)B_{m,j}\big(\psi ^{2l}_y\big)+\\ & \quad +\delta _1^{-2}A_{m,k}\big(\psi ^{s-2l+2}_x\big)B_{m,j}\big(\psi ^{2l}_y\big)+ \delta _2^{-2}A_{m,k}\big(\psi ^{s-2l}_x\big)B_{m,j}\big(\psi ^{2l+2}_y\big)+\\ & \quad +\delta _1^{-2}\delta _2^{-2}A_{m,k} \big(\psi ^{s-2l+2}_x\big)B_{m,j}\big(\psi ^{2l+2}_y\big)\Big] \sum ^s_{i=0}\tbinom {s}{i}\omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}; \delta _1, \delta _2\right)\! , \end{align*}

\begin{align*} & |(R_{m,m} f)(x,y)|\leq \\ & \leq \tfrac {1}{s!}m^{s-\gamma _s}\sum ^{\tfrac {s}{2}}_{l=0} \Bigg[\tfrac {(T_{m,s-2l}L_m)(x)}{m^{s-2l}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{2l}}+\\ & \quad +\delta _1^{-2}\tfrac {(T_{m,s\! -\! 2l+2}L_m)(x)}{m^{s\! -\! 2l+2}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{2l}}\! +\! \delta _2^{-2}\tfrac {(T_{m,s\! -\! 2l}L_m)(x)}{m^{s\! -\! 2l}}\, \tfrac {(T_{m,2l\! +\! 2}K_m)(y)}{m^{2l+2}}\! +\\ & \quad +\delta _1^{-2}\delta _2^{-2}\tfrac {(T_{m,s-2l+2}L_m(x)}{m^{s-2l+2}}\, \tfrac {(T_{m,2l+2}K_m)(y)}{m^{2l+2}}\Bigg]\! \sum ^s_{i=0}\tbinom {s}{i}\omega _{total}\! \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i};\delta _1, \delta _2\right)\! , \end{align*}

\begin{align*} & |(R_{m,m}f)(x,y)|\leq \\ & \leq \tfrac {1}{s!}\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l} \Bigg[\tfrac {(T_{m,s-2l}L_m)(x)}{m^{\alpha _{s-2l}}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\, m^{\alpha _{s-2l}+\beta _{2l}-\gamma _s}+\\ & \quad +\delta _1^{-2}\tfrac {(T_{m,s-2l+2}L_m)(x)}{m^{\alpha _{s-2l+2}}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\, m^{\alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2}+\\ & \quad +\delta _2^{-2}\tfrac {(T_{m,s-2l} L_m)(x)}{m^{\alpha _{s-2l}}}\, \tfrac {(T_{m,2l+2}K_m)(y)}{m^{\beta _{2l+2}}}\, m^{\alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2}+\\ & \quad +\delta _1^{-2}\delta _2^{-2}\tfrac {(T_{m, s-2l+2}L_m)(x)}{m^{\alpha _{s-2l+2}}}\, \tfrac {(T_{m,2l+2}K_m)(y)}{m^{\beta _{2l+2}}}\, m^{\alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4} \Bigg]\cdot \\ & \quad \cdot \sum ^s_{i=0}\tbinom {s}{i}\omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i};\delta _1, \delta _2\right)\! . \end{align*}

We have $\delta _s\leq -(\alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2)$, $\delta _s\leq -(\alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2)$, $\delta _s\leq -(\alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4)$ for any $l\in \bigg\{ 0,1,\dots , \tfrac {s}{2}\bigg\} $, from where $\delta _s+\alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2\leq 0$, $\delta _s+\alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2\leq 0$, $2\delta _s+\alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4\leq 0$, for any $l\in \bigg\{ 0,1,\dots , \tfrac {s}{2}\bigg\} $. From the relation (9) and from the inequalities above, we have

\begin{align*} m^{\alpha _{s-2l}+\beta _{2l}-\gamma _s}& \leq 1,\\ m^{\delta _s+\alpha _{s-2l+2}+\beta _{2l}-\gamma _s-2}& \leq 1\\ m^{\delta _s+\alpha _{s-2l}+\beta _{2l+2}-\gamma _s-2}& \leq 1,\\ m^{2\delta _s+\alpha _{s-2l+2}+\beta _{2l+2}-\gamma _s-4}& \leq 1, \end{align*}

where $l\in \bigg\{ 0,1,\dots , \tfrac {s}{2}\bigg\} $.

Considering $\delta _1=\delta _2=\tfrac {1}{\sqrt{m^{\delta _s}}}$ , we have

\begin{align} \label{3.15} & |(R_{m,m}f)(x,y)|\leq \\ & \leq \tfrac {1}{s!}\sum ^{\tfrac {s}{2}}_{l=0} \tbinom {\tfrac {s}{2}}{l}\Bigg[\tfrac {(T_{m,s-2l}L_m)(x)}{m^{\alpha _{s-2l}}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}+\\ & \quad +\! \tfrac {(T_{m, s-2l+2}L_m)(x)}{m^{\alpha _{s-2l+2}}}\, \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\! +\! \tfrac {(T_{m,s-2l}L_m)(x)}{m^{\alpha _{s-2l}}} \tfrac {(T_{m,2l+2}K_m)(y)}{m^{\beta _{2l+2}}}\nonumber \\ & \quad +\tfrac {(T_{m,s-2l+2}L_m)(x)}{m^{\alpha _{s-2l+2}}}\, \tfrac {(T_{m,2l+2}K_m)(y)}{m^{\beta _{2l+2}}}\Bigg]\cdot \nonumber \\ & \quad \cdot \sum ^s_{i=0}\tbinom {s}{i}\omega _{total} \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ; \tfrac {1}{\sqrt{m^{\delta _s}}}\, ,\tfrac {1}{\sqrt{m^{\delta _s}}}\right)\! .\nonumber \end{align}

Taking (7), (8) into account and considering the fact that

\[ \lim _{m\to \infty }\omega _{total} \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ; \tfrac {1}{\sqrt{m^{\delta _s}}}\, ,\tfrac {1}{\sqrt{m^{\delta _s}}}\right)= \omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ;0,0\right)=0, \]

$i\in \{ 0,1,\dots ,s\} $, from (34) we have that

\begin{equation} \label{3.16} \lim _{m\to \infty }(R_{m,m}f)(x,y)=0. \end{equation}

From (28) and (36), (23) follows.

If in addition (24), (25) take place, then (34) becomes

\begin{align} \label{3.17} |(R_{m,m}f)(x,y)|& \leq \tfrac {1}{s!}\sum ^{\tfrac {s}{2}}_{l=0} \tbinom {\tfrac {s}{2}}{l}(a_{s-2l}+a_{s-2l+2})(b_{2l}+b_{2l+2})\cdot \\ & \quad \cdot \sum ^s_{i=0}\tbinom {s}{i}\omega _{total} \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ; \tfrac {1}{\sqrt{m^{\delta _s}}}\, ,\tfrac {1}{\sqrt{m^{\delta _s}}}\right)\nonumber \end{align}

for any $m\in \mathbb N$, $m\geq m(s)$ and for any $(x,y)\in K_1\times K_2$, from which, the convergence from (22) is uniform on $K_1\times K_2$. From (28) and (37), (25) follows.

Proof â–¼

Corollary 3.3

Let $f:I_1\times I_2\to \mathbb R$ be a bivariate function.
If $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$ and $f$ is continuous in $(x,y)$, then

\begin{equation} \label{3.18} \lim _{m\to \infty }\left(L^\ast _{m,m}f\right)(x,y)=f(x,y). \end{equation}

If $f$ is continuous on $(I_1\cap J_1)\times (I_2\cap J_2)$ and there exist the intervals $K_1\subset I_1\cap J_1$, $K_2\subset I_2\cap J_2$ such that there exist $m(0)\in \mathbb N$ and $a_2, b_2\in \mathbb R$ depending on $K_1$, respectively $K_2$, so that for any $m\in \mathbb N$, $m\geq m(0)$ and any $(x,y)\in K_1\times K_2$ we have

\begin{equation} \label{3.19} \tfrac {(T_{m,2}L_m)(x)}{m^{\alpha _2}}\leq a_2, \end{equation}

\begin{equation} \label{3.20} \tfrac {(T_{m,2}K_m)(y)}{m^{\beta _2}}\leq b_2, \end{equation}

then the convergence given in $(\ref{3.18})$ is uniform on $K_1\times K_2$ and

\begin{equation} \label{3.21} \left|\left(L^\ast _{m,m}f\right)(x,y)\! -\! f(x,y)\right|\! \leq \! (1+a_2)(1+b_2)\omega _{total}\left(f; \tfrac {1}{\sqrt{m^{\delta _0}}}\, ,\tfrac {1}{\sqrt{m^{\delta _0}}}\right) \end{equation}

for any $(x,y)\in K_1\times K_2$, any $m\in \mathbb N$, $m\geq m(0)$.

Proof â–¼

It result from Theorem 3.2 for $s=0$ and one verifies immediately that $\alpha _0\! =\! \beta _0\! =\! \gamma _0=0$, $a_0\! =\! b_0\! =\! 1$, $\delta _0\! =\! -\max \bigg\{ \beta _2\! -\! 2, \alpha _2\! -\! 2, \tfrac {1}{2}\, (\alpha _2\! +\! \beta _2\! -\! 4)\bigg\} $.

In the following, in addition we suppose that

\begin{equation} \label{3.22} \alpha _{s+2}<\alpha _s+2,\; \beta _{s+2}<\beta _s+2 \end{equation}

and for any $f\in E(I_1\times I_2)$ we have

\begin{equation} \label{3.23} A_{m,n,k,j}(f_x)=B_{n,j}(f_x), \end{equation}

\begin{equation} \label{3.24} A_{m,n,k,j}(f^y)=A_{m,k}(f^y), \end{equation}

\begin{equation} \label{3.25} A_{m,n,k,j}(f)=A_{m,k}(B_{n,j}(f_x))=B_{n,j}(A_{m,k}(f^y)) \end{equation}

for any $x\in I_1$, $y\in I_2$, $k\in \{ 0,1,\dots , p_m\} \cap \mathbb N_0$, $j\in \{ 0,1,\dots ,q_n\} \cap \mathbb N_0$, $m,n\in \mathbb N$.

In [9] are given the following results, where if $P_m=L_m$, $m\in \mathbb N$ then $I=I_1$, $J=J_1$, $\eta _j=\alpha _j$, $k_j=a_j$, $j\in \{ s, s+2\} $ and if $P_m=K_m$, $m\in \mathbb N$ then $I=I_2$, $J=J_2$, $\eta _j=\beta _j$, $k_j=b_j$, $j\in \{ s, s+2\} $.

Theorem 3.4

Let $f:I\to \mathbb R$ be a function. If $x\in I\cap J$ and $f$ is a $s$ times differentiable in $x$ with $f^{(s)}$ continuous in $x$, then

\begin{equation} \label{3.26} \lim _{m\to \infty } m^{s-\eta _s} \left[(P_m f)(x)-\sum ^s_{i=0}\tfrac {f^{(i)}(x)}{m^i i!} (T_{m,i}P_m)(x)\right]=0. \end{equation}

Assume that $f$ is $s$ times differentiable function on $I$, with $f^{(s)}$ continuous in $I$ and there exists an interval $M\subset I\cap J$ such that there exist $m(s)\in \mathbb N$ and $k_j\in \mathbb R$ depending on $M$ so that for any $m\geq m(s)$ and any $x\in M$ we have

\begin{equation} \label{3.27} \tfrac {(T_{m,j}P_m)(x)}{m^{\eta _j}}\leq k_j \end{equation}

where $j\in \{ s, s+2\} $. Then the convergence given in $(\ref{3.26})$ is uniform on $M$ and

\begin{align} \label{3.28} m^{s-\eta _s}\Big|(P_m f)(x)\! -\! \sum ^s_{i=0} \tfrac {f^{(i)}(x)}{m^i i!}\, (T_{m,i}P_m)(x)\Big| \! \leq \! \tfrac {1}{s!}\, (k_s\! +\! k_{s+2})\omega \left(f^{(s)}; \tfrac {1}{\sqrt{m^{2\! +\! \eta _s\! -\! \eta _{s+2}}}}\right)\nonumber \end{align}

for any $x\in M$ and $m\geq m(s)$.

Now, let $\left(UL^\ast _{m,m}\right)_{m,n\geq 1}$ be the GBS operators associated to the $\left(L^\ast _{m,n}\right)_{m,n\geq 1}$ operators.

Lemma 3.5

If $m,n\in \mathbb N$, then $UL^\ast _{m,n}$ have the form

\begin{equation} \label{3.29} \left(UL^\ast _{m,n} f\right)(x,y)=(K_n f_x)(y)+ (L_m f^y)(x)-\left(L_{m,n}^\ast f\right)(x,y) \end{equation}

for any $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$, any $f\in E(I_1\times I_2)$.

Proof â–¼

We have

\begin{align*} \left(UL^\ast _{m,n} f\right)(x,y) & =\left(L^\ast _{m,n}(f(x, \ast )+f(\cdot , y)-f(\cdot , \ast )\right)(x,y)\\ & =\left(L^\ast _{m,n} f(x, \ast )\right)\! (x,y)\! +\! \left(L^\ast _{m,n} f(\cdot , y)\right)\! (x,y)\! -\! \left(L^\ast _{m,n} f\right)\! (x,y)\\ & =\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x)\psi _{n,j}(y) A_{m,n,k,j}(f_x)+\\ & \quad +\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x) \psi _{n,j}(y)A_{m,n,k,j}(f^y)-\left(L^\ast _{m,n} f\right)(x,y) \end{align*}

and taking (44), (45) into account, we obtain

\begin{align*} \left(UL^\ast _{m,n} f\right)(x,y) & =\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x)\psi _{n,j}(y) B_{n,j}(f_x)+\\ & \quad +\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x)\psi _{n,j}(y) A_{m,k}(f^y)-\left(L^\ast _{m,n} f\right)(x,y)\\ & =\left(\sum ^{p_m}_{k=0}\varphi _{m,k}(x)\right)\! \left(\sum ^{q_n}_{j=0}\psi _{n,j}(y)B_{n,j}(f_x)\right)+\\ & \quad +\left(\sum ^{p_m}_{k=0}\varphi _{m,k}(x)A_{m,k} (f^y)\right)\! \left(\sum ^{q_n}_{j=0}\psi _{n,j}(y)\right)\! -\left(L^\ast _{m,n}f\right)(x,y). \end{align*}

From (5), (6), (15) and (16), the relation (48) is obtained.

Proof â–¼

Theorem 3.6

\begin{align} \label{3.30} & \lim _{m\to \infty }m^{s-\gamma _s}\Bigg\{ \left(UL^\ast _{m,m}f\right) (x,y)-\\ & \quad -\sum ^s_{i=0} \tfrac {1}{m^i i!}\bigg[\Big(\tfrac {\partial ^i f}{\partial \tau ^i}(x,y) \left(T_{m,i}K_m\right)(y)+\tfrac {\partial ^i f}{\partial t^i}(x,y)\left(T_{m,i}L_m\right)(x)\Big)-\nonumber \\ & \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l} \partial \tau ^l} (x,y)\left(T_{m,i-l}L_m\right)(x)\left(T_{m,l} K_m\right)(y)\bigg]\Bigg\} =0.\nonumber \end{align}

\begin{equation} \label{3.31} \tfrac {\left(T_{m,2l}L_m\right)(x)}{m^{\alpha _{2l}}} \leq a_{2l}, \end{equation}

\begin{equation} \label{3.32} \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l} \end{equation}

where $l\in \Big\{ 0,1,\dots , \tfrac {s}{2}+1\Big\} $, then the convergence given in $(\ref{3.30})$ is uniform on $K_1\times K_2$ and

\begin{align} \label{3.33} & m^{s-\gamma _s}\Bigg|(UL^\ast _{m,m} f)(x,y)-\\ & \quad \quad \quad -\sum ^s_{i=0}\tfrac {1}{m^i i!}\Bigg[\tfrac {\partial ^i f}{\partial \tau ^i} (x,y)(T_{m,i}K_m)(y)+ \tfrac {\partial ^i f}{\partial t^i}(x,y)(T_{m,i}L_m)(x)-\nonumber \\ & \quad \quad \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^i} (x,y)(T_{m,i-l}L_m)(x)(T_{m,l}K_m)(y)\Bigg]\Bigg|\leq \nonumber \\ & \leq \! \tfrac {1}{s!}\! \Bigg[\! (b_s\! +\! b_{s+2}) \omega \Big(\tfrac {\partial ^s f_x}{\partial \tau ^s};\tfrac {1} {\sqrt{m^{2+\beta _s-\beta _{s+2}}}}\Big)\! +\! (a_s\! +\! a_{s+2})\omega \Big(\tfrac {\partial ^s f^y}{\partial t^s};\! \tfrac {1} {\sqrt{m^{2+\alpha _s-\alpha _{s+2}}}}\Big)\! +\nonumber \\ & \quad \quad \quad +\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l}(a_{s-2l}+a_{s-2l+2}) (b_{2l}+b_{2l+2}) \sum ^s_{i=0}\tbinom {s}{i}\cdot \nonumber \\ & \quad \quad \quad \cdot \omega _{total}\left(\! \tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ;\tfrac {1}{\sqrt{m^{\delta _s}}}\, , \tfrac {1}{\sqrt{m^{\delta _s}}}\! \right)\! \Bigg]\nonumber \end{align}

for any $(x,y)\in K_1\times K_2$, any $m\in \mathbb N$, $m\geq m(s)$, where $\delta _s$ is given in $(\ref{3.9})$.

Proof â–¼

From (9), it results that $\gamma _s\geq \alpha _s$, $\gamma _s\geq \beta _s$, and then $s-\gamma _s\leq s-\alpha _s$ and $s-\gamma _s\leq s-\beta _s$. We use the (46) relation from Theorem 3.4 for the functions $f_x$ and $K_n$, $n\in \mathbb N$ operators and for the function $f^y$ and $L_m$, $m\geq 1$ operators, the (22) relation from Theorem 3.2 for the function $f$ and then we obtain the (49) relation. If we note by $S$ the left member of (52) relation, we can write

\begin{align*} S& =m^{s-\gamma _s}\, \Bigg|\Big[(K_m f_x)(y)- \sum ^s_{i=0}\tfrac {1}{m^i i!}\, \tfrac {\partial ^i f}{\partial \tau ^i} \, (x,y)\left(T_{m,i}K_m\right)(y)\Big]+\\ & \quad +\Big[(L_m f^y)(x)-\sum ^s_{i=0}\tfrac {1}{m^i i!}\, \tfrac {\partial ^i f}{\partial t^i}\, (x,y)\left(T_{m,i}L_m\right)(x)\Big]+\\ & \quad +\Big[\sum ^s_{i=0}\tfrac {1}{m^i i!}\sum ^i_{l=0}\tbinom {i}{l} \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y) \left(T_{m,i-l}L_m\right)(x)\left(T_{m,l} K_m\right)(y)-\\ & \quad -\left(L^\ast _{m,m} f\right)(x,y)\Big]\! \Bigg|\! \leq \end{align*}

\begin{align*} & \leq m^{s-\alpha _s}\Big|\! \left(L_m f_x\right)(y)\! -\! \sum ^s_{i=0}\tfrac {1}{m^i i!}\, \tfrac {\partial ^i f}{\partial \tau ^i} (x,y)\left(T_{m,i}K_m\right)\! (y)\! \Big|+\\ & \quad +m^{s-\gamma _s}\Big|(L_m f^y)(x)-\sum ^s_{i=0}\tfrac {1}{m^i i!} \tfrac {\partial ^i f}{\partial t^i}\, (x,y)\left(T_{m,i}L_m\right)(x)\Big|+\\ & \quad + m^{s-\gamma _s}\Big| \left(L^\ast _{m,m}f\right)(x,y)-\sum ^s_{i=0}\tfrac {1}{m^i i!} \sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}\, (x,y)\cdot \\ & \quad \cdot \left(T_{m,i-l}L_m\right)(x) \left(T_{m,l}K_m\right)(y)\Big| \end{align*}

and taking Theorem 3.4 and relation (25) into account we obtain the first inequality from (52). From (52) the uniform convergence for (49) results.

Proof â–¼

Corollary 3.7

Let $f:I_1\times I_2\to \mathbb R$ be a bivariate function.

If $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$ and $f$ is continuous in $(x,y)$, then

\begin{equation} \label{3.34} \lim _{m\to \infty }\left(UL^\ast _{m,m} f\right)(x,y)=f(x,y). \end{equation}

Assume that $f$ is continuous on $(I_1\cap J_1)\times (I_2\cap J_2)$ and there exist the intervals $K_1\subset I_1\cap J_1$, $K_2\subset I_2\cap J_2$ such that there exist $m(0)\in \mathbb N$ and $a_2, b_2\in \mathbb R$ depending on $K_1$, respectively $K_2$ so that for any $m\in \mathbb N$, $m\geq m(0)$ and any $x\in K_1$, $y\in K_2$, we have

\begin{equation} \label{3.35} \tfrac {\left(T_{m,2}L_m\right)(x)}{m^{\alpha _2}} \leq a_2 \end{equation}

and

\begin{equation} \label{3.36} \tfrac {(T_{m,2}K_m)(y)}{m^{\beta _2}}\leq b_2. \end{equation}

Then the convergence given in $(\ref{3.34})$ is uniform on $K_1\times K_2$ and

\begin{align} \label{3.37} & \left|\left(UL^\ast _{m,m} f\right)(x,y)-f(x,y)\right|\leq \\ & \leq (1+b_2)\omega \left(f_x;\tfrac {1}{\sqrt{m^{2-\beta _2}}}\right)+ (1+a_2)\omega \left(f^y;\tfrac {1}{\sqrt{m^{2-\alpha _2}}}\right)+\nonumber \\ & \quad +(1\! +\! a_2)(1\! +\! b_2)\omega _{total}\left(\! f; \tfrac {1}{\sqrt{m^{\delta _0}}}\, , \tfrac {1}{\sqrt{m^{\delta _0}}}\! \right)\! \leq \! (1\! +\! b_2)\omega \left(\! f_x; \tfrac {1}{\sqrt{m^{\delta _0}}}\! \right)+\nonumber \\ & \quad +(1\! +\! a_2)\omega \left(\! f^y;\tfrac {1}{\sqrt{m^{\delta _0}}}\! \right)\! +\! (1\! +\! a_2)(1\! +\! b_2)\omega _{total}\left(\! f; \tfrac {1}{\sqrt{m^{\delta _0}}}\, ,\tfrac {1}{\sqrt{m^{\delta _0}}}\! \right)\nonumber \end{align}

for any $(x,y)\in K_1\times K_2$ and any $m\in \mathbb N$, $m\geq m(0)$, where

\[ \delta _0=-\max \bigg\{ \beta _2-2, \alpha _2-2, \tfrac {1}{2}\, (\alpha _2+\beta _2-4)\bigg\} . \]

Proof â–¼

It results from Theorem 3.6 for $s=0$.

Proof â–¼

Because every application is a simple substitute in the results of this section, we won’t replace anything.

Application 3.8

If $I_1=J_1=[0,1]$, $I_2=J_2=[0,\infty )$, $E_1(I_1)=C([0,1])$, $E_2(I_2)\! =\! C_2([0,\infty ))$, $p_m\! =\! m$, $q_n\! =\! \infty $, $\varphi _{m,k}(x)\! =\! p_{m,k}(x)\! =\! \tbinom {m}{k}x^k(1-x)^{m-k}$,
$\psi _{n,j}(y)={\rm e}^{-ny}\tfrac {(ny)^j}{j!}$ , $A_{m,k}(f^y)=f\left(\tfrac {k}{m}\, , y\right)$, $B_{n,j}(f_x)=f\left(x, \tfrac {j}{n}\right)$,
$A_{m,n,j,k}(f)=f\left(\tfrac {k}{m}\, ,\tfrac {j}{n}\right)$ for any $(x,y)\in [0,1]\times [0,\infty )$, $m,n\in \mathbb N$,
$k\in \{ 0,1,\dots , m\} $, $j\in \mathbb N_0$ and $f\in E([0,1]\times [0,\infty ))$. Then starting from $(B_m)_{m\geq 1}$ the Bernstein operators and $(S_n)_{n\geq 1}$ the Mirakjan-Favard-Szász operators, we obtain the operators $\left(L^\ast _{m,n}\right)_{m,n\geq 1}$ and $\left(UL^\ast _{m,n}\right)_{m,n\geq 1}$ defined for any function $f\in E([0,1]\times [0,\infty ))$, any $(x,y)\in [0,1]\times [0,\infty )$ and $m,n\in \mathbb N$ by

\begin{equation} \label{3.38} \left(L^\ast _{m,n}f\right)(x,y)= \sum ^m_{k=0}\sum ^\infty _{j=0}\tbinom {m}{k}x^k(1-x)^{m-k}{\rm e}^{-ny} \tfrac {(ny)^j}{j!}\, f\left(\tfrac {k}{m}\, ,\tfrac {j}{n}\right), \end{equation}
57

\begin{equation} \label{3.39} \left(UL^\ast _{m,n}f\right)(x,y)=(S_n f_x)(y)+ (B_m f^y)(x)-\left(L^\ast _{m,n}f\right)(x,y). \end{equation}
58

In this case $\alpha _2=\beta _2=1$, $\delta _0=1$, $K_1=[0,1]$, $K_2=[0,b]$, $b{\gt}0$, $a_2=\tfrac {5}{4}$ , $b_2=b$ and $m(0)=1$ (see [3]).

Application 3.9

If $I_1=J_1=[0,\infty )$, $I_2\! =\! J_2\! =\! [0,1]$, $E_1(I_1)\! =\! C_2([0,\infty ))$,
$E_2(I_2)\! =\! L_1([0,1])$, $p_m\! =\! \infty $, $q_n=n$, $\varphi _{m,k}(x)\! =\! (1+x)^{-m} \tbinom {m\! +\! k\! -\! 1}{k}\big(\tfrac {x}{1\! +\! x}\big)^k$, $\psi _{n,j}(y)\! =\! \tbinom {n}{j}y^j(1-y)^{n\! -\! j}$,

\begin{align*} A_{m,k}(f^y)\! =& \! f\! \left(\! \tfrac {k}{m}\, ,y\! \right), \\ B_{n,j}(f_x)\! =& \! (n\! +\! 1)\! \displaystyle \int ^{\tfrac {j\! +\! 1}{n\! +\! 1}}_{\tfrac {j}{n\! +\! 1}} \! f(x,t){\rm d}t, \\ A_{m,n,k,j}(f)=& (n+1)\displaystyle \int ^{\tfrac {j+1}{n+1}}_{\tfrac {j}{n+1}} f\left(\tfrac {k}{m}\, , t\right){\rm d}t, \end{align*}

for any $(x,y)\in [0,\infty )\times [0,1]$, $m,n\in \mathbb N$, $k\in \mathbb N_0$, $j\in \{ 0,1,\dots , n\} $ and $f\in E([0,\infty )\times [0,1])$. Then with the aid of $(V_m)_{m\geq 1}$ Baskakov operators and $(K_n)_{n\geq 1}$ Kantorovich operators, we obtain the operators $\left(L^\ast _{m,n}\right)_{m,n\geq 1}$ and $\left(UL^\ast _{m,n}\right)_{m,n\geq 1}$ defined for any function $f\in E([0,\infty )\times [0,1])$, $(x,y)\in [0,\infty )\times [0,1]$ and $m,n\in \mathbb N$ by
\begin{align} \label{3.40} & \left(L^\ast _{m,n}f\right)(x,y)= \\ & =\sum ^\infty _{k=0} \sum ^n_{j=0}(n\! +\! 1)\tbinom {m\! +\! k\! -\! 1}{k}\tbinom {n}{j}(1\! +\! x)^{-m} \left(\! \tfrac {x}{1\! +\! x}\! \right)^k y^j(1-y)^{n-j}\int ^{\tfrac {j+1}{n+1}}_{\tfrac {j}{n+1}} f\left(\tfrac {k}{m}\, , t\right){\rm d}t,\nonumber \end{align}

\begin{equation} \label{3.41} \left(UL^\ast _{m,n}f\right)(x,y)=(K_n f_x)(y)+ (V_m f^y)(x)-\left(L^\ast _{m,n}f\right)(x,y). \end{equation}
60

In this case $\alpha _2=\beta _2=1$, $\delta _0=1$, $K_1=[0,b]$, $b{\gt}0$, $K_2=[0,1]$, $a_2=b(b+1)$, $b_2=1$ and $m(0)=3$ (see [2] and [3]).

Application 3.10

If $I_1=I_2=J_1=J_2=[0,1]$, $E_1(I_1)=C([0,1])$, $E_2(I_2)=L_1([0,1])$, $p_m=\infty $, $q_n=n$, $\varphi _{m,k}(x)=\tbinom {m+k}{k}(1-x)^{m+1}x^k$, $\psi _{n,j}(y)=p_{n,j}(y)$,

\begin{align*} A_{m,k}(f^y)=& f\left(\tfrac {k}{m+k}\, , y\right), \\ B_{n,j}(f_x)=& (n+1)\displaystyle \int ^1_0p_{n,j}(t) f(x,t){\rm d}t,\\ A_{m,n,k,j}(f)=& (n+1)\displaystyle \int ^1_0 p_{n,j}(t)f\left(\tfrac {k}{m+k}\, , t\right){\rm d}t \end{align*}

for any $(x,y)\in [0,1]\times [0,1]$, $m,n\in \mathbb N$, $k\in \mathbb N_0$, $j\in \{ 0,1,\dots , n\} $ and $f\in E([0,1]\times [0,1])$. Then with $(Z_m)_{m\geq 1}$ the Meyer-König and Zeller operators and $(M_n)_{n\geq 1}$ the Durrmeyer operators, we construct the operators $\left(L^\ast _{m,n}\right)_{m,n\geq 1}$ and $\left(UL^\ast _{m,n}\right)_{m,n\geq 1}$ defined for any function $f\in E([0,1]\times [0,1])$, any $(x,y)\in [0,1]\times [0,1]$ and $m,n\in \mathbb N$ by
\begin{align} \label{3.42} & \left(L^\ast _{m,n}f\right)(x,y) = \\ & =\sum ^\infty _{k=0} \sum ^n_{j=0}(n+1)\tbinom {m+k}{k}\tbinom {n}{j} (1\! -\! x)^{m+1}x^k y^j(1\! -\! y)^{n-j}\! \int ^1_0\! p_{n,j}(t)f\left(\! \tfrac {k}{m\! +\! k}\, , t\! \right) {\rm d}t,\nonumber \end{align}

\begin{equation} \label{3.43} \left(UL^\ast _{m,n} f\right)(x,y)= (M_n f_x)(y)+(Z_m f^y)(x)-\left(L^\ast _{m,n}f\right)(x,y). \end{equation}
62

In this case $\alpha _2=\beta _2=1$, $\delta _0=1$, $K_1=K_2=[0,1]$, $a_2=2$, $b_2=\tfrac {3}{2}$ and $m(0)=3$ (see [2] and [3]).

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