<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>The approximation of bivariate functions by modified bivariate operators and GBS operators associated: The approximation of bivariate functions by modified bivariate operators and GBS operators associated</title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">

<div class="titlepage">
<h1>The approximation of bivariate functions by modified bivariate operators and GBS operators associated</h1>
<p class="authors">
<span class="author">Ovidiu T. Pop\(^\ast \)</span>
</p>
<p class="date">May 7, 2012</p>
</div>
<p>\(^\ast \) National College “Mihai Eminescu", 5 Mihai Eminescu St., 440014 Satu Mare, Romania, e-mail: <span class="tt">ovidiutiberiu@yahoo.com</span> </p>

<div class="abstract"><p> In this paper we demonstrate a Voronovskaja-type theorem and approximation theorem for a class of modified operators and <i class="it">Generalized Boolean Sum</i> (GBS) associated operators obtained (see (<a href="#1.3">3</a>)) from given operators. </p>
<p><b class="bf">MSC.</b> 41A10, 41A25, 41A35, 41A36, 41A63 </p>
<p><b class="bf">Keywords.</b> Linear positive operators, bivariate operators, GBS operators, <br />Voronovskaja-type theorem, approximation theorem. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>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. </p>
<p>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. </p>
<p>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. </p>
<p>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 \(x\in I\). </p>
<p>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 </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> and then \(\omega (-f, \delta )=\omega (f, \delta )\). </p>
<p>If \(n\in \mathbb N\), \(\delta \geq 0\) and \(f_1, f_2,\dots , f_n\in B(I)\), then </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>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. </p>
<p>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 <br />\(f\in E(I_1\times I_2)\), any \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\) by </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> 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]). </p>
<p>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 \(s\in I_1\). </p>
<p>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 </p>
<div class="displaymath" id="1.4">
  \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}
</div>
<p> is called the first order modulus of smoothness of function \(f\) or total modulus of continuity of function \(f\) (see [10]). </p>
<p>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 </p>
<div class="equation" id="1.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.5} (T_i L_m)(x)=m^i\left(L_m\psi ^i_x\right)(x) \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> 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 \(J\). </p>
<p>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 </p>
<div class="equation" id="1.6">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> and </p>
<div class="equation" id="1.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.7} \sum ^k_{j=0}\tbinom {m}{j}\tbinom {n}{k-j}= \tbinom {m+n}{k} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> are known, where \(k,m,n\in \mathbb N_0\). </p>
<h1 id="a0000000003">2 Preliminaries</h1>
<p>In this section we recall the construction and the results from the paper [8]. </p>
<p>Let \(p_m=m\) for any \(m\in \mathbb N\) or \(p_m=\infty \) for any \(m\in \mathbb N\) and similarly is defined \(q_n\), \(n\in \mathbb N\). </p>
<p>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 \(A_{m,k}:E(I_1)\to \mathbb R\), \(B_{n,j}:E_2(I_2)\to \mathbb R\). </p>
<p>For \(m,n\in \mathbb N\) define the sequences of operators \((L_m)_{m\geq 1}\) and \((K_n)_{n\geq 1}\) by </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} (L_m f)(x)=\sum ^{p_m}_{k=0}\varphi _{m,k}(x) A_{m,k}(f), \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<div class="equation" id="2.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.2} (K_n g)(y)=\sum ^{q_n}_{j=0}\psi _{n,j}(y)B_{n,j}(g) \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> 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 \(I_2\). </p>
<p>The operators \((L_m)_{m\geq 1}\) and \((K_n)_{n\geq 1}\) are linear and positive on <br />\(E_1(I_1\cap J_1)\), respectively \(E_2(I_2\cap J_2)\). </p>
<p>In the following let \(s\in \mathbb N_0\), \(s\) even. </p>
<p>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 </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} \lim _{m\to \infty }\tfrac {(T_j L_m)(x)}{m^{\alpha _j}}= \alpha _j(x) \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> for any \(x\in I_1\cap J_1\), </p>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} \lim _{n\to \infty }\tfrac {(T_j K_n)(y)}{n^{\beta _j}}=b_j(y) \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> for any \(y\in I_2\cap J_2\) and if we note </p>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> then </p>
<div class="equation" id="2.6">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> where \(l\in \{ 0,1,2,\dots \tfrac {s}{2}\} \). </p>
<p>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 \(y\in I_2\big\} \). </p>
<p>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 </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> 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 \(x\in I_1\), \(y\in I_2\). </p>
<p>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 </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> is named the bivariate operator of \(LK\)-type. </p>
<p>The operators \(\left(L_{m,n}^\ast \right)_{m,n\geq 1}\) are linear and positive on \(E\left((I_1\! \cap \!  J_1)\! \times \!  (I_2\! \cap \! J_2)\right)\). </p>
<p>In the following we consider that </p>
<div class="equation" id="2.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.9} (T_{0}L_m)(x)=A_{m,0}(e_0)=1 \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> for any \(x\in I_1\cap J_1\), \(m\in \mathbb N\) and </p>
<div class="equation" id="2.10">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.10} (T_{0}K_n)(y)=B_{n,0}(e_0)=1 \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> for any \(y\in I_2\cap J_2\), \(n\in \mathbb N\). </p>
<p>From (<a href="#2.9">16</a>), (<a href="#2.10">17</a>) it results immediately that </p>
<div class="equation" id="2.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.11} \sum ^{p_m}_{k=0}\varphi _{m,k}(x)=1 \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> for any \(x\in I_1\cap J_1\), \(m\in \mathbb N\), </p>
<div class="equation" id="2.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.12} \sum ^{q_n}_{j=0}\psi _{n,j}(y)=1 \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> for any \(y\in I_2\cap J_2\), \(n\in \mathbb N\) and then </p>
<div class="equation" id="2.13">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.13} \alpha _0=\beta _0=0. \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>In the following, we note by \(t, \tau \) the first, respectively second variable of function. <div class="theorem_thmwrapper " id="th_2.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> 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 </p>
<div class="displaymath" id="2.14">
  \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}
</div>
<p> 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 </p>
<div class="equation" id="2.15">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.15} \tfrac {(T_{2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<div class="equation" id="2.16">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.16} \tfrac {(T_{2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l}, \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p> 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 </p>
<div class="displaymath" id="2.17">
  \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}
</div>
<p> for any \((x,y)\in K_1\times K_2\), any \(m\in \mathbb N\) with \(m\geq m(s)\), where </p>
<div class="displaymath" id="2.18">
  \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}
</div>

  </div>
</div> </p>
<p>In the following, in addition we suppose that </p>
<div class="equation" id="2.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.19} \alpha _{s+2}<\alpha _s+2, \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<div class="equation" id="2.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.20} \beta _{s+2}<\beta _s+2 \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> and for any \(f\in E(I_1\times I_2)\) we have </p>
<div class="equation" id="2.21">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.21} A_{m,n,k,j}(f_x)=B_{n,j}(f_x), \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<div class="equation" id="2.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.22} A_{m,n,k,j}(f^y)=A_{m,k}(f^y), \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<div class="equation" id="2.23">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> for any \(x\in I_1\), \(y\in I_2\), \(k\in \! \{ 0,1,\dots , p_m\} \cap \,  \mathbb N_0\), \(j\in \{ 0,1,\dots ,q_n\} \cap \, \mathbb N_0\), \(m,n\in \mathbb N\). </p>
<p>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. </p>
<p><div class="lemma_thmwrapper " id="a0000000004">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>If \(m,n\in \mathbb N\), then \(UL^\ast _{m,n}\) have the form </p>
<div class="equation" id="2.24">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> for any \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\), any \(f\in E(I_1\times I_2)\). </p>

  </div>
</div> <div class="theorem_thmwrapper " id="th_2.2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> 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 </p>
<div class="displaymath" id="2.25">
  \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}
</div>
<p> 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 </p>
<div class="equation" id="2.26">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.26} \tfrac {\left(T_{2l}L_m\right)(x)}{m^{\alpha _{2l}}} \leq a_{2l}, \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<div class="equation" id="2.27">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.27} \tfrac {(T_{2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l} \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p> 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 </p>
<div class="displaymath" id="2.28">
  \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}
</div>
<p> 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 \((\ref{2.18})\). </p>

  </div>
</div> </p>
<h1 id="a0000000005">3 Main results</h1>
<p>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. </p>
<p>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 \(E(I_1\times I_2)\). </p>
<p>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 </p>
<div class="displaymath" id="3.1">
  \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}
</div>
<div class="equation" id="3.2">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p> for any \((t,\tau )\in I_2\times I_2\) and \(f\in E^r(I_1\times I_2)\). </p>
<p><div class="definition_thmwrapper " id="a0000000006">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">4</span>
  </div>
  <div class="definition_thmcontent">
  <p><em>For \(m,n\in \mathbb N\) define the operator \(L_{m,n,r}\) for any <br />\(f\in E^r(I_1\times I_2)\), any \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\) by <div class="displaymath" id="3.3">
  \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}
</div> </em></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000007">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1</span>
  </div>
  <div class="remark_thmcontent">
  <p><em>If in (<a href="#2.8">15</a>) 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 (<a href="#3.3">38</a>).<span class="qed">â–¡</span></em></p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000008">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2</span>
  </div>
  <div class="remark_thmcontent">
  <p><em>The operators \((L_{m,n,r})_{m,n\geq 1}\) are construct about G. H. Kirov idea, but for the bivariate functions.<span class="qed">â–¡</span></em></p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000009">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3</span>
  </div>
  <div class="remark_thmcontent">
  <p><em>For \(r=0\) we obtain \(L_{m,n,0}=L^\ast _{m,n}\) for any \(m,n\in \mathbb N\).<span class="qed">â–¡</span></em></p>

  </div>
</div> <div class="theorem_thmwrapper " id="th_3.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> a) Let \(f:I_1\times I_2\to \mathbb R\) be a function, \(f\in E^r(I_1\times I_2)\). <br />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 </p>
<div class="equation" id="3.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.4} \lim _{m\to \infty }(L_{m,m,r}f)(x,y)=f(x,y), \end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<div class="equation" id="3.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.5} \lim _{m\to \infty }(UL_{m,m,r}f)(x,y)=f(x,y) \end{equation}
  </div>
  <span class="equation_label">40</span>
</p>
</div>
<p> if \(s=0\) and </p>
<div class="displaymath" id="3.6">
  \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}
</div>
<p> and </p>
<div class="displaymath" id="3.7">
  \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}
</div>
<p> 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 </p>
<div class="equation" id="3.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.8} \tfrac {(T_{2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}
  </div>
  <span class="equation_label">43</span>
</p>
</div>
<div class="equation" id="3.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.9} \tfrac {(T_{2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l}, \end{equation}
  </div>
  <span class="equation_label">44</span>
</p>
</div>
<p> where \(l\in \left\{ 0,1,\dots , \tfrac {s}{2}+1\right\} \).<br />c) In hypothesis b), then the convergence given in \((\ref{3.4})--(\ref{3.7})\) are uniform on \(K_1\times K_2\) and </p>
<div class="equation" id="3.10">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">45</span>
</p>
</div>
<div class="displaymath" id="3.11">
  \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}
</div>
<p> for any \((x,y)\in K_1\times K_2\), any \(m\in \mathbb N\), \(m\geq m(0)\) if \(s=0\), where </p>
<div class="equation" id="3.12">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">47</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="3.13">
  \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}
</div>
<div class="displaymath" id="3.14">
  \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}
</div>
<p> 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 \((\ref{2.18})\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For \((t,\tau )\in I_1\times I_2\) and \(\alpha \in \{ 0,1,\dots , r\} \), we have from (<a href="#3.1">36</a>) </p>
<div class="displaymath" id="a0000000011">
  \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*}
</div>
<p> from where </p>
<div class="displaymath" id="a0000000012">
  \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*}
</div>
<p> 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 </p>
<div class="displaymath" id="a0000000013">
  \[  \tfrac {\partial ^i g_{x,y,\alpha }}{\partial t^{i-l}\partial \tau ^l}(x,y)= (-1)^\alpha \sum ^\alpha _{\beta =0}\tbinom {\alpha }{\beta } \tbinom {i-l}{\alpha -\beta }\tbinom {l}{\beta }(\alpha -\beta )!\beta !\tfrac {\partial ^i f} {\partial t^{i-l}\partial \tau ^l}(x,y)  \]
</div>
<p> so </p>
<div class="equation" id="3.15">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p> for \(i\in \{ 0,1,\dots , s\} \). </p>
<p>If \(i\in \{ 0,1,\dots , s\} \) and \(l=i\), from (<a href="#3.15">50</a>) we have that </p>
<div class="displaymath" id="a0000000014">
  \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*}
</div>
<p> But \(\displaystyle \sum ^r_{\alpha =0}(-1)^\alpha \tbinom {i}{\alpha }= (-1)^r\tbinom {i-1}{r}\), then </p>
<div class="equation" id="3.16">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">51</span>
</p>
</div>
<p> and similarly </p>
<div class="equation" id="3.17">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">52</span>
</p>
</div>
<p> \(i\in \{ 1,\dots , s\} \). </p>
<p>If \(i=0\) then </p>
<div class="equation" id="3.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.18} g(x,y)=f(x,y). \end{equation}
  </div>
  <span class="equation_label">53</span>
</p>
</div>
<p>Applying Theorem <a href="#th_2.1">1</a> and Theorem <a href="#th_2.2">3</a> for the function \(g_{x,y}\) and taking (<a href="#1.6">6</a>), (<a href="#1.7">7</a>) and (<a href="#3.15">50</a>)–(<a href="#3.18">53</a>) into account, we obtain the assertions from Theorem <a href="#th_3.1">5</a>. <div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000016">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(m,n\in \mathbb N\), then \(UL_{m,n,r}\) have the form </p>
<div class="displaymath" id="3.19">
  \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}
</div>
<p> for any \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\) and \(f\in E^r(I_1\times I_2)\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> From (<a href="#3.1">36</a>) 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 </p>
<div class="displaymath" id="a0000000018">
  \[  g_{x,y}(x, \tau )=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} g_{x,y,\alpha }(x,\tau )=\sum ^r_{\alpha =0}\tfrac {1}{\alpha !} \tfrac {\partial ^\alpha f_x}{\partial \tau ^\alpha }\, (\tau ).  \]
</div>
<p> 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 (<a href="#2.24">31</a>), we obtain relation (<a href="#3.19">54</a>). <div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, we give an application.<br />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 <br />\((L_{m,n,r})_{m,n\geq 1}\) 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 </p>
<div class="displaymath" id="3.20">
  \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}
</div>
<div class="displaymath" id="3.21">
  \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}
</div>
<p> 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]). </p>
<p><div class="theorem_thmwrapper " id="a0000000020">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f:[0,1]\times [0,\infty )\) be a function, \(f\in E^r([0,1]\times [0,\infty ))\).<br />If \((x,y)\in [0,1]\times [0,\infty ))\) and \(f\) is continuous in \((x,y)\), then </p>
<div class="equation" id="3.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.22} \lim _{m\to \infty }(L_{m,n,r}f)(x,y)=f(x,y) \end{equation}
  </div>
  <span class="equation_label">57</span>
</p>
</div>
<p> and </p>
<div class="equation" id="3.23">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.23} \lim _{m\to \infty }(UL_{m,n,r}f)(x,y)=f(x,y). \end{equation}
  </div>
  <span class="equation_label">58</span>
</p>
</div>
<p>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 </p>
<div class="displaymath" id="3.24">
  \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}
</div>
<div class="displaymath" id="3.25">
  \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}
</div>
<p> for any \((x,y)\in [0,1]\times [0,b]\), any \(m\in \mathbb N\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> It results from Theorem <a href="#th_3.1">5</a>. <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><i class="sc">C. Badea</i> and <i class="sc">C. Cottin</i>, <em>Korovkin-type Theorems for Generalized Boolean Sum Operators</em>, Colloquia Mathematica Societatis János Bolyai, <b class="bfseries">58</b>, Approximation Theory, Kecskemét (Hungary), 1990, pp.&#160;51–67. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">D. Bărbosu</i>, <em>Polynomial Approximation by Means of Schurer-Stancu type Operators</em>, Editura Universităţii de Nord Baia Mare, 2006. </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><i class="sc">G.H. Kirov</i>, <em>A generalization of the Bernstein polynomials</em>, Math. Balkanica, New Series, <b class="bfseries">6</b> (1992) no. 2, pp.&#160;147–153. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no1-art9"> <i class="sc">O.T. Pop</i>, <em>The generalization of Voronovskaja’s theorem for a class of linear and positive operators</em>, Rev. Anal. Numér. Théor. Approx., <b class="bfseries">34</b> (2005) no. 1, pp.&#160;79–91. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">O.T. Pop</i>, <em>About some linear and positive operators defined by infinite sum</em>, Dem. Math., <b class="bfseries">XXXIX</b> (2006) no. 2, pp.&#160;377–388. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><i class="sc">O.T. Pop</i>, <em>The generalization of Voronovskaja’s theorem for a class of bivariate operators</em>, Studia Univ. “Babeş-Bolyai", Mathematica <b class="bfseries">LIII</b> (2008) no. 2, pp.&#160;85–107. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><i class="sc">O.T. Pop</i>, <em>The generalization of Voronovskaja’s theorem for a class of bivariate operators defined by infinite sum</em>, Anal. Univ. Oradea, Fasc. Matematica, <b class="bfseries">XV</b> (2008), pp.&#160;155–169. </p>
</dd>
  <dt><a name="8">8</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2011-vol40-no1-art7"> <i class="sc">O.T. Pop</i>, <em>The approximation of bivariate functions by bivariate operators and GBS operators</em>, Rev. Anal. Numé. Théor. Approx., <b class="bfseries">40</b> (2011) no. 1, pp.&#160;64–79. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="9">9</a></dt>
  <dd><p><i class="sc">O.T. Pop</i>, <em>About some linear and positive operators</em>, International Journal of Mathematics and Mathematical Sciences, <b class="bf">2007</b>, Article ID91781, 2007, 13 pages. </p>
</dd>
  <dt><a name="10">10</a></dt>
  <dd><p><i class="sc">O.T. Pop</i>, <em>Voronovskaja-type theorems and approximation theorems for a class of GBS operators</em>, Fasciculi Mathematici, <b class="bfseries">42</b> (2009), pp.&#160;91–108. </p>
</dd>
  <dt><a name="11">11</a></dt>
  <dd><p><i class="sc">A.F. Timan</i>, <em>Theory of Approximation of Functions of Real Variable</em>, New York: Macmillan Co. 1963, MR22#8257. </p>
</dd>
  <dt><a name="12">12</a></dt>
  <dd><p><i class="sc">E. Voronovskaja</i>, <em>Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein</em>, C. R. Acad. Sci. URSS, 1932, pp.&#160;79–85. </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>