The approximation of bivariate functions by modified bivariate operators and GBS operators associated

Ovidiu T. Pop$^\ast $

May 7, 2012

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

In this paper we demonstrate a Voronovskaja-type theorem and approximation theorem for a class of modified operators and Generalized Boolean Sum (GBS) associated operators obtained (see (3)) from given operators.

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

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

1 Introduction

In this paper we start from a class of linear and positive operators. We construct the bivariate operators and GBS operators associated for bivariate functions.

The aim of this paper is to modify these operators applying the G.H. Kirov idea (see [3]) and for the new class of operators, we demonstrate a Voronovskaja-type theorem and an approximation theorem.

Let $\mathbb N$ be the set of positive integers and $\mathbb N_0=\mathbb N\cup \{ 0\} $. In this section we recall some notions and results which we will use in this paper.

We consider $I\subset \mathbb R$, $I$ an interval and we shall use the following function sets: $B(I)=\big\{ f|f:I\to \mathbb R$, $f$ bounded on $I\big\} $, $(C(I)=\big\{ f|f:I\to \mathbb R$, $f$ continuous on $I\big\} $ and $C_B(I)=B(I)\cap C(I)$. For $x\in I$, let the function $\psi _x:I\to \mathbb R$, $\psi _x(t)=t-x$ for any $t\in I$ and let $e_0:I\to \mathbb R$, $e_0(x)=1$ 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 \big\{ |f(x')-f(x'')|:x', x''\in I, |x'-x''|\leq \delta \big\} \end{equation}

and then $\omega (-f, \delta )=\omega (f, \delta )$.

If $n\in \mathbb N$, $\delta \geq 0$ and $f_1, f_2,\dots , f_n\in B(I)$, then

\begin{equation} \label{1.2} \omega (f_1+f_2+\dots +f_n; \delta )\leq \omega (f_1; \delta )+\omega (f_2; \delta )+\dots + \omega (f_n;\delta ). \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 $J_1\times J_2$. In these sets, we have the classical operations of addition of functions and multiplication with scalars of functions.

Let $L:E(I_1\times I_2)\to F(J_1\times J_2)$ 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{1.3} (UL f)(x,y)=\left(L\left((x, \ast )+f(\cdot , y)-f(\cdot , \ast )\right)\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{1.4} \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 [10]).

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

\begin{equation} \label{1.5} (T_i L_m)(x)=m^i\left(L_m\psi ^i_x\right)(x) \end{equation}

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

We consider that if $n\in \mathbb N_0$ and $k\in \mathbb Z\backslash \{ 0,1,\dots , n\} $, then $\tbinom {n}{k}=0$ and $\tbinom {0}{0}=1$. The identities

\begin{equation} \label{1.6} \sum ^k_{j=0}(-1)^j\tbinom {n}{j}= \left\{ \begin{array}{ll} (-1)^k\tbinom {n-1}{k}, & 0\leq k\leq n-1,\; n\in \mathbb N\\ 1, & n=0\\ 0, & \text{in other cases} \end{array}\right. \end{equation}

and

\begin{equation} \label{1.7} \sum ^k_{j=0}\tbinom {m}{j}\tbinom {n}{k-j}= \tbinom {m+n}{k} \end{equation}

are known, where $k,m,n\in \mathbb N_0$.

2 Preliminaries

In this section we recall the construction and the results from the paper [8].

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 $A_{m,k}:E(I_1)\to \mathbb R$, $B_{n,j}:E_2(I_2)\to \mathbb R$.

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}

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

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 $I_2$.

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

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_j L_m)(x)}{m^{\alpha _j}}= \alpha _j(x) \end{equation}

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

\begin{equation} \label{2.4} \lim _{n\to \infty }\tfrac {(T_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 \{ 0,1,\dots , \tfrac {s}{2}\} \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 \{ 0,1,2,\dots \tfrac {s}{2}\} $.

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 $x\in I_1$, $y\in I_2$.

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}

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

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

In the following we consider that

\begin{equation} \label{2.9} (T_{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_{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 (16), (17) it results immediately that

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

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

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

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

\begin{equation} \label{2.13} \alpha _0=\beta _0=0. \end{equation}

In the following, we note by $t, \tau $ the first, respectively second variable of function.

Theorem 1

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{2.14} & \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_{i-l}L_m\right)(x) \left(T_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{2.15} \tfrac {(T_{2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}

\begin{equation} \label{2.16} \tfrac {(T_{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 $(\ref{2.14})$ is uniform on $K_1\times K_2$ and

\begin{align} \label{2.17} & 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_{i-l}L_m\right)(x)\left(T_{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 \quad \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{2.18} \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 \{ 0,1,\dots ,\tfrac {s}{2} \} \! \bigg\} .\nonumber \end{align}

In the following, in addition we suppose that

\begin{equation} \label{2.19} \alpha _{s+2}<\alpha _s+2, \end{equation}

\begin{equation} \label{2.20} \beta _{s+2}<\beta _s+2 \end{equation}

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

\begin{equation} \label{2.21} A_{m,n,k,j}(f_x)=B_{n,j}(f_x), \end{equation}

\begin{equation} \label{2.22} A_{m,n,k,j}(f^y)=A_{m,k}(f^y), \end{equation}

\begin{equation} \label{2.23} 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 $x\in I_1$, $y\in I_2$, $k\in \! \{ 0,1,\dots , p_m\} \cap \, \mathbb N_0$, $j\in \{ 0,1,\dots ,q_n\} \cap \, \mathbb N_0$, $m,n\in \mathbb N$.

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

Lemma 2

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

\begin{equation} \label{2.24} \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 $f\in E(I_1\times I_2)$.

Theorem 3

\begin{align} \label{2.25} & \lim _{m\to \infty }m^{s-\gamma _s}\Bigg\{ \left(UL^\ast _{m,m}f\right) (x,y)-\sum ^s_{i=0} \tfrac {1}{m^i i!}\Bigg[\Bigg(\tfrac {\partial ^i f}{\partial \tau ^i}(x,y) \left(T_i K_m\right)(y)\\ & \quad \quad \quad \quad \quad +\tfrac {\partial ^i f}{\partial t^i}(x,y)\left(T_i L_m\right)(x)\Bigg)-\nonumber \\ & \quad \quad \quad \quad \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l} \partial \tau ^l} (x,y)\left(T_{i-l}L_m\right)(x)\left(T_l K_m\right)(y)\Bigg]\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 any $(x,y)\in K_1\times K_2$ we have

\begin{equation} \label{2.26} \tfrac {\left(T_{2l}L_m\right)(x)}{m^{\alpha _{2l}}} \leq a_{2l}, \end{equation}

\begin{equation} \label{2.27} \tfrac {(T_{2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l} \end{equation}

where $l\in \{ 0,1,\dots , \tfrac {s}{2}+1\} $, then the convergence given in $(\ref{2.25})$ is uniform on $K_1\times K_2$ and

\begin{align} \label{2.28} & m^{s-\gamma _s}\Bigg|(UL^\ast _{m,m} f)(x,y)-\\ & \quad -\sum ^s_{i=0}\tfrac {1}{m^i i!}\Bigg[\tfrac {\partial ^i f}{\partial \tau ^i} (x,y)(T_i K_m)(y)+ \tfrac {\partial ^i f}{\partial t^i}(x,y)(T_i L_m)(x)-\nonumber \\ & \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l} (x,y)(T_{i-l}L_m)(x)(T_l K_m)(y)\Bigg]\Bigg|\leq \nonumber \\ & \leq \! \tfrac {1}{s!}\! \Bigg[\! (b_s\! +\! b_{s+2}) \omega \! \left(\! \tfrac {\partial ^s f_x}{\partial \tau ^s};\! \tfrac {1} {\sqrt{m^{2+\beta _s-\beta _{s+2}}}}\! \right)\! +\! (a_s\! +\! a_{s+2})\omega \left(\! \tfrac {\partial ^s f^y}{\partial t^s};\! \tfrac {1} {\sqrt{m^{2+\alpha _s-\alpha _{s+2}}}}\! \right)\! +\nonumber \\ & \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 \! \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 $(\ref{2.18})$.

3 Main results

Let the fixed number $r\in \mathbb N_0$ and $\left(L^\ast _{m,n}\right)_{m,n\geq 1}$ the operators which we construct in Preliminaries.

We denote by $E^r(I_1\times I_2)$ the set of all function $f\in E(I_1\times I_2)$ with the partial derivatives $\tfrac {\partial ^k f}{\partial t^{k-i}\partial \tau ^i}$ , $i\in \{ 0,1,\dots , k\} $ and $k\in \{ 0,1,\dots , r\} $ belonging to $E(I_1\times I_2)$.

For $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$, $\alpha \in \{ 0,1,\dots , r\} $ we consider the functions $g_{x,y,\alpha }$, $g_{x,y}:I_1\times I_2\to \mathbb R$, defined by

\begin{align} \label{3.1} g_{x,y,\alpha }(t,\tau ) & =\left(\tfrac {\partial }{\partial t}\, (x-t)+ \tfrac {\partial }{\partial \tau }\, (y-\tau )\right)^\alpha f(t,\tau )=\\ & =\sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta }\tfrac {\partial ^\alpha f} {\partial t^{\alpha -\beta }\partial \tau ^\beta }\, (t,\tau )(x-t)^{\alpha -\beta } (y-\tau )^\beta ,\nonumber \end{align}

\begin{equation} \label{3.2} g_{x,y}(t,\tau )=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !}\, g_{x,y,\alpha }(t,\tau ) \end{equation}

for any $(t,\tau )\in I_2\times I_2$ and $f\in E^r(I_1\times I_2)$.

Definition 4

For $m,n\in \mathbb N$ define the operator $L_{m,n,r}$ for any
$f\in E^r(I_1\times I_2)$, any $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$ by

\begin{align} \label{3.3} (L_{m,n,r}f)(x,y)& =\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0} \varphi _{m,k}(x)\psi _{n,j}(y)\sum ^r_{\alpha =0}\tfrac {1}{\alpha !}\, A_{m,n,k,j}(g_{x,y,\alpha })=\\ & =\sum ^{p_m}_{k=0}\sum ^{q_n}_{j=0}\varphi _{m,k}(x) \psi _{n,j}(y)\sum ^r_{\alpha =0}\sum ^\alpha _{\beta =0} \tfrac {1}{\alpha !}\tbinom {\alpha }{\beta }\cdot \nonumber \\ & \quad \cdot A_{m,n,k,j} \left(\tfrac {\partial ^\alpha f}{\partial t^{\alpha -\beta } \partial \tau ^\beta }(x-\cdot )^{\alpha -\beta } (y-\ast )^\beta \right).\nonumber \end{align}

Remark 1

If in (15) we replace the function $f$ by the function $g_{x,y}$, $(x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)$, we obtain the operators from (38).â–¡

Remark 2

The operators $(L_{m,n,r})_{m,n\geq 1}$ are construct about G. H. Kirov idea, but for the bivariate functions.â–¡

Remark 3

For $r=0$ we obtain $L_{m,n,0}=L^\ast _{m,n}$ for any $m,n\in \mathbb N$.â–¡

Theorem 5

a) Let $f:I_1\times I_2\to \mathbb R$ be a function, $f\in E^r(I_1\times I_2)$.
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{equation} \label{3.4} \lim _{m\to \infty }(L_{m,m,r}f)(x,y)=f(x,y), \end{equation}

\begin{equation} \label{3.5} \lim _{m\to \infty }(UL_{m,m,r}f)(x,y)=f(x,y) \end{equation}

if $s=0$ and

\begin{align} \label{3.6} & \lim _{m\to \infty }m^{s-\gamma _s}\Bigg[(L_{m,m,r}f)(x,y)-f(x,y)-\\ & \quad -\! (-1)^r\! \sum ^s_{i=1}\tfrac {1}{m^i i!}\tbinom {i\! -\! 1}{r}\sum ^i_{l=0} \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}\, (x,y) (T_{i-l}L_m)(x)(T_l K_m)(y)\! \Bigg]\! \! =\! 0\nonumber \end{align}

and

\begin{align} \label{3.7} & \lim _{m\to \infty }m^{s-\gamma _s}\Bigg\{ (UL_{m,m,r}f)(x,y)-f(x,y)-\\ & \quad -(-1)^r\sum ^s_{i=1}\tfrac {1}{m^ii!}\tbinom {i\! -\! 1}{r}\! \Bigg[\! \Bigg(\! \tfrac {\partial ^i f}{\partial \tau ^i}(x,y)(T_i K_m)(y)\! +\! \tfrac {\partial ^i f}{\partial t^i}(x,y)(T_i L_m)(x)\! \! \Bigg)\! -\nonumber \\ & \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l} (x,y)(T_{i-l}L_m)(x)(T_l K_m)(y)\bigg]\Bigg\} =0.\nonumber \end{align}

b) Assume that $f$ 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{3.8} \tfrac {(T_{2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}

\begin{equation} \label{3.9} \tfrac {(T_{2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l}, \end{equation}

where $l\in \left\{ 0,1,\dots , \tfrac {s}{2}+1\right\} $.
c) In hypothesis b), then the convergence given in $(\ref{3.4})--(\ref{3.7})$ are uniform on $K_1\times K_2$ and

\begin{equation} \label{3.10} \left|(L_{m,m,r}f)(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}

\begin{align} \label{3.11} & \left|(UL_{m,m,r}f)(x,y)-f(x,y)\right|\leq (1+b_2)\omega \left(f_x; \tfrac {1}{\sqrt{m^{2-\beta _2}}}\right)+\\ & \quad + (1\! +\! a_2)\left(\! f^y; \tfrac {1}{\sqrt{m^{2-\alpha _2}}}\! \right) +(1\! +\! a_2)(1\! +\! b_2)\omega _{total}\left(\! f; \tfrac {1}{\sqrt{m^{\delta _0}}}\, , \tfrac {1}{\sqrt{m^{\delta _0}}}\! \right)\! \leq \nonumber \\ & \leq (1+b_2)\omega \left(f_x; \tfrac {1}{\sqrt{m^{\delta _0}}}\right)+(1+a_2) \omega \left(f^y; \tfrac {1}{\sqrt{m^{\delta _0}}}\right)+\nonumber \\ & \quad +(1+a_2)(1+b_2)\omega \left(f; \tfrac {1}{\sqrt{m^{\delta _0}}}\, , \tfrac {1}{\sqrt{m^{\delta _0}}}\right)\nonumber \end{align}

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

\begin{equation} \label{3.12} \delta _0=-\max \left\{ \beta _2-2, \alpha _2-2, \tfrac {1}{2} (\alpha _2+\beta _2-4)\right\} \end{equation}

and

\begin{align} \label{3.13} & m^{s-\gamma _s}\Bigg|(L_{m,m,r}f)(x,y)- f(x,y)-\\ & \quad -(-1)^r\sum ^s_{i=1}\tfrac {1}{m^i i!}\tbinom {i-1}{r} \sum ^i_{l=0}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y) (T_{i-l}L_m)(x)(T_l K_m)(y)\Bigg|\leq \nonumber \\ & \leq \tfrac {1}{s!}\sum ^r_{\alpha =0}\tbinom {s}{\alpha }\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 \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),\nonumber \end{align}

\begin{align} \label{3.14} & m^{s-\gamma _s}\Bigg|(UL_{m,m,r}f)(x,y)-f(x,y)-\\ & \quad -(-1)^r\sum ^s_{i=1}\tfrac {1}{m^i i!}\tbinom {i-1}{r}\Bigg[\Bigg( \tfrac {\partial ^i f}{\partial \tau ^i}(x,y)(T_i K_m)(y)+ \tfrac {\partial ^i f}{\partial t^i}(x,y)(T_i L_m)(x)\Bigg)-\nonumber \\ & \quad -\sum ^i_{l=0}\tbinom {i}{l}\tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y)(T_{i-l}L_m)(x)(T_l K_m)(y)\Bigg]\Bigg|\leq \nonumber \\ & \leq \tfrac {1}{s!}\Bigg\{ \tbinom {s-1}{r}\Bigg[(b_s+b_{s+2})\omega \left(\tfrac {\partial ^s f_x}{\partial \tau ^s};\tfrac {1}{\sqrt{m^{2+\beta _s-\beta _{s+2}}}} \right)+\nonumber \\ & \quad +(a_s+a_{s+2})\omega \left(\tfrac {\partial ^s f^y}{\partial t^s}; \tfrac {1}{\sqrt{m^{2+a_s-a_{s+2}}}}\right)\Bigg]+\sum ^s_{\alpha =0} \tbinom {s}{\alpha }\sum ^{\tfrac {s}{2}}_{l=0}\tbinom {\tfrac {s}{2}}{l}\cdot \nonumber \\ & \quad \cdot (a_{s-2l}+a_{s-2l+2})(b_{2l}+b_{2l+2})\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)\! \! \Bigg\} \nonumber \end{align}

for any $(x,y)\in K_1\times K_2$, any $m\in \mathbb N$, $m\geq m(s)$, if $s\geq 2$, where $\delta _s$ is given in $(\ref{2.18})$.

Proof â–¼

For $(t,\tau )\in I_1\times I_2$ and $\alpha \in \{ 0,1,\dots , r\} $, we have from (36)

\begin{gather*} \tfrac {\partial ^i g_{x,y,\alpha }}{\partial t^{i-l}\partial \tau ^l} (t,\tau )\! =\! \sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta } \tfrac {\partial ^i}{\partial t^{i-l}\partial \tau ^l} \left(\! \tfrac {\partial ^\alpha f}{\partial t^{\alpha -\beta }\partial \tau ^\beta } (t,\tau )(x\! -\! t)^{\alpha -\beta }(y\! -\! \tau )^\beta \! \right)\! =\\ =\sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta }\tfrac {\partial ^l}{\partial \tau ^l} \Bigg[\sum ^{i-l}_{i_1=0}\tbinom {i\! -\! l}{i_1} \tfrac {\partial ^{\alpha +i-l-i_1}f}{\partial t^{\alpha -\beta +i-l-i_1}\partial \tau ^\beta } (t,\tau )((x\! -\! t)^{\alpha -\beta })^{(i_1)}(y\! -\! \tau )^\beta \Bigg], \end{gather*}

from where

\begin{align*} \tfrac {\partial ^i g_{x,y,\alpha }}{\partial t^{i-l}\partial \tau ^l}(t, \tau )& = \sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta }\sum ^{i-l}_{i_1=0} \sum ^l_{i_2=0}\tbinom {i-l}{i_1}\tbinom {l}{i_2}\cdot \\ & \quad \cdot \tfrac {\partial ^{\alpha +i-i_1-i_2}f}{\partial t^{\alpha -\beta +i-l-i_1} \partial \tau ^{\beta +l-i_2}}(t,\tau )((x-t)^{\alpha -\beta })^{(i_1)} ((y-\tau )^\beta )^{(i_2)}. \end{align*}

Taking into account that the function $((x-t)^{\alpha -\beta })^{(i_1)}((y-\tau )^\beta )^{(i_2)}$ of variable $(t, \tau )$ takes nonzero value in $(x,y)$ if and only if $\alpha -\beta =i_1$ and $\beta =i_2$, we have that

\[ \tfrac {\partial ^i g_{x,y,\alpha }}{\partial t^{i-l}\partial \tau ^l}(x,y)= (-1)^\alpha \sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta } \tbinom {i-l}{\alpha -\beta }\tbinom {l}{\beta }(\alpha -\beta )!\beta !\tfrac {\partial ^i f} {\partial t^{i-l}\partial \tau ^l}(x,y) \]

\begin{equation} \label{3.15} \tfrac {\partial ^i g_{x,y}}{\partial t^{i-l}\partial \tau ^l}(x,y)= \left(\sum ^r_{\alpha =0}\sum ^\alpha _{\beta =0}(-1)^\alpha \tbinom {i-l}{\alpha -\beta }\tbinom {l}{\beta }\right) \tfrac {\partial ^i f}{\partial t^{i-l}\partial \tau ^l}(x,y) \end{equation}

for $i\in \{ 0,1,\dots , s\} $.

If $i\in \{ 0,1,\dots , s\} $ and $l=i$, from (50) we have that

\begin{align*} \tfrac {\partial ^i g_{x,y}}{\partial \tau ^i}(x,y)& =\sum ^r_{\alpha =0} \sum ^\alpha _{\beta =0}(-1)^\alpha \tbinom {0}{\alpha -\beta }\tbinom {i}{\beta } \tfrac {\partial ^i f}{\partial \tau ^i}(x,y)=\\ & =\sum ^r_{\alpha =0}(-1)^\alpha \tbinom {i}{\alpha }\tfrac {\partial ^i f} {\partial \tau ^i}(x,y). \end{align*}

But $\displaystyle \sum ^r_{\alpha =0}(-1)^\alpha \tbinom {i}{\alpha }= (-1)^r\tbinom {i-1}{r}$, then

\begin{equation} \label{3.16} \tfrac {\partial ^i g_{x,y}}{\partial \tau ^i}\, (x,y)= (-1)^r\tbinom {i-1}{r}\tfrac {\partial ^i f}{\partial \tau ^i}\, (x,y) \end{equation}

and similarly

\begin{equation} \label{3.17} \tfrac {\partial ^i g_{x,y}}{\partial t^i}\, (x,y)=(-1)^r \tbinom {i-1}{r}\tfrac {\partial ^i f}{\partial t^i}\, (x,y), \end{equation}

$i\in \{ 1,\dots , s\} $.

If $i=0$ then

\begin{equation} \label{3.18} g(x,y)=f(x,y). \end{equation}

Applying Theorem 1 and Theorem 3 for the function $g_{x,y}$ and taking (6), (7) and (50)–(53) into account, we obtain the assertions from Theorem 5.

Proof â–¼

Theorem 6

If $m,n\in \mathbb N$, then $UL_{m,n,r}$ have the form

\begin{align} \label{3.19} (UL_{m,n,r} f)(x,y) & =\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} \Bigg[\left(K_n\tfrac {\partial ^\alpha f_x}{\partial \tau ^\alpha }(y-\ast )^\alpha \right)(y)+\\ & \quad +\left(L_m\tfrac {\partial ^\alpha f^y}{\partial t^\alpha }(x-\cdot )^\alpha \right)(x)\Bigg]- (L_{m,n,r}f)(x,y)\nonumber \end{align}

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

Proof â–¼

From (36) we have that $g_{x,y,\alpha }(x,\tau )=\tfrac {\partial ^\alpha f}{\partial \tau ^\alpha }\, (x, \tau )$, so $g_{x,y,\alpha }(x,\tau )=\tfrac {\partial ^\alpha f_x}{\partial \tau ^\alpha }\, (\tau )$ and then

\[ g_{x,y}(x, \tau )=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} g_{x,y,\alpha }(x,\tau )=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} \tfrac {\partial ^\alpha f_x}{\partial \tau ^\alpha }\, (\tau ). \]

Similarly $g_{x,y}(t,y)=\displaystyle \sum ^r_{\alpha =0}\tfrac {1}{\alpha !} \tfrac {\partial ^\alpha f^y}{\partial t^\alpha }\, (t)$ and if we replace the function $f$ by the function $g_{x,y}$ in (31), we obtain relation (54).

Proof â–¼

Now, we give an application.
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 $, $\varphi _{m,k}(x)=p_{m,k}(x)=\tbinom {m}{k}x^k(1-x)^{m-k}$, $\psi _{n,j}(y)=e^{-ny}\tfrac {(ny)^j}{j!}$ , $A_{m,k}(f^y)=f\left(\tfrac {k}{m}\, , y\right)$, $B_{n,j}(f_x)=f\left(x, \tfrac {j}{n}\right)$, $A_{m,n,k,j}(f)=f\left(\tfrac {k}{m}\, , \tfrac {j}{n}\right)$ 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 with $r\in \mathbb N_0$, $(B_m)_{m\geq 1}$ the Bernstein operators and $(S_n)_{n\geq 1}$ the Mirakjan-Favard-Szász operators, we obtain the operators
$(L_{m,n,r})_{m,n\geq 1}$ and $(UL_{m,n,r})_{m,n\geq 1}$ defined for any function $f\in E([0,1]\times [0,\infty ))$, $(x,y)\in [0,1)\times [0,\infty )$ and $m,n\in \mathbb N$ by

\begin{align} \label{3.20} & (L_{m,n,r}f)(x)=\sum ^m_{k=0}\sum ^\infty _{j=0} \tbinom {m}{k}x^k(1-x)^{m-k}e^{-ny}\tfrac {(ny)^j}{j!}\, \cdot \\ & \quad \cdot \sum ^r_{\alpha =0}\tfrac {1}{\alpha !}\sum ^\alpha _{\beta =0} \tbinom {\alpha }{\beta }\tfrac {\partial ^\alpha f}{\partial t^{\alpha -\beta } \partial \tau ^\beta }\left(\tfrac {k}{m}\, ,\tfrac {j}{n}\right) \left(x-\tfrac {k}{m}\right)^{\alpha -\beta } \left(y-\tfrac {j}{n}\right)^\beta \! ,\nonumber \end{align}

\begin{align} \label{3.21} & (UL_{m,n,r}f)(x,y)=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} \Bigg[e^{-ny}\sum ^\infty _{j=0}\tfrac {(ny)^j}{j!}\, \tfrac {\partial ^\alpha f_x}{\partial \tau ^\alpha }\, \left(y-\tfrac {j}{n}\right)^\alpha +\\ & \quad +\sum ^m_{k=0}\tbinom {m}{k}x^k(1-x)^{m-k} \tfrac {\partial ^\alpha f^y}{\partial t^\alpha } \left(x-\tfrac {k}{m}\right)^\alpha \Bigg]-(L_{m,n,r}f)(x,y).\nonumber \end{align}

We have $\alpha _2=\beta _2=1$, $\delta _0=1$, $K_1=[0,1]$, $K_2=[0,b]$, $b{\gt}0$, $a_2=\tfrac {5}{4}$ , $b_2=b$ and $m(0)=1$ (see [8]).

Theorem 7

Let $f:[0,1]\times [0,\infty )$ be a function, $f\in E^r([0,1]\times [0,\infty ))$.
If $(x,y)\in [0,1]\times [0,\infty ))$ and $f$ is continuous in $(x,y)$, then

\begin{equation} \label{3.22} \lim _{m\to \infty }(L_{m,n,r}f)(x,y)=f(x,y) \end{equation}

and

\begin{equation} \label{3.23} \lim _{m\to \infty }(UL_{m,n,r}f)(x,y)=f(x,y). \end{equation}

If $f$ is continuous in $[0,1]\times [0,\infty )$, then the convergence given in $(\ref{3.22})$, $(\ref{3.23})$ are uniform on $[0,1]\times [0,b]$, $b{\gt}0$ and

\begin{align} \label{3.24} \left|(L_{m,n,r}f)(x,y)-f(x,y)\right|\leq \tfrac {9}{4}\, (1+b)\omega _{total}\left(f; \tfrac {1}{\sqrt{m}}\, ,\tfrac {1}{\sqrt{m}}\right)\! , \end{align}

\begin{align} \label{3.25} \big|(UL_{m,n,r}f)(x,y)-f(x,y)\big|& \leq (1+b)\omega \left(f_x; \tfrac {1}{\sqrt{m}}\right)+ \tfrac {9}{4}\omega \left(f^y; \tfrac {1}{\sqrt{m}}\right)\\ & \quad +\tfrac {9}{4}\, (1+b)\omega _{total}\left(f; \tfrac {1}{\sqrt{m}}\, , \tfrac {1}{\sqrt{m}}\right)\nonumber \end{align}

for any $(x,y)\in [0,1]\times [0,b]$, any $m\in \mathbb N$.

Proof â–¼

It results from Theorem 5.

Proof â–¼

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