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<title>The approximation of bivariate functions by bivariate operators and GBS operators: The approximation of bivariate functions by bivariate operators and GBS operators</title>
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<h1>The approximation of bivariate functions by bivariate operators and GBS operators</h1>
<p class="authors">
<span class="author">Ovidiu T. Pop\(^\ast \)</span>
</p>
<p class="date">June 6, 2010.</p>
</div>
<p>\(^\ast \)National College “Mihai Eminescu", 5 Mihai Eminescu Street, 440014 Satu Mare,<br />Romania, e-mail: <span class="tt">ovidiutiberiu@yahoo.com</span> </p>

<div class="abstract"><p> In this paper we demonstrate a general approximation theorems for the bivariate functions by bivariate operators and GBS (Generalized Boolean Sum) 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, Voronovskaja-type theorem, approximation theorem. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>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. </p>
<p>The aim of this paper is to demonstrate a general approximation theorem of bivariate functions by special bivariate operators. </p>
<p>Let \(\mathbb N\) be the set of positive integers and \(\mathbb N_0=\mathbb N\cup \{ 0\} \). </p>
<p>In this section we recall some notions which we will use in this paper. </p>
<p>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 \(x\in I\). </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 \left\{ |f(x')-f(x'')|: x',\;  x'\in I,\; |x'-x''|\leq \delta \right\} \! . \end{equation}
  </div>
  <span class="equation_label">1</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 \(J_1\times J_2\) 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 </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">2</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="1.3">
  \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}
</div>
<p> is called the first order modulus of smoothness of function \(f\) or total modulus of continuity of function \(f\) (see [11]). </p>
<p>The first order modulus of smoothness for bivariate functions has properties similar to the properties of the first modulus of smoothness for univariate functions. </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 \(m\in \mathbb N\) and \(i\in \mathbb N_0\) define \(T_{m,i}\) by </p>
<div class="equation" id="1.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.4} (T_{m,i}L_m)(x)=m^i\left(L_m \psi ^i_x\right)(x) \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> for any \(x\in I\cap J\), where \(E(I)\), \(F(J)\) are subsets of the set of real functions defined on \(I\), respectively \(J\). </p>
<h1 id="a0000000003">2 Preliminaries</h1>
<p> 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 \(q_n\), \(n\in \mathbb N\). </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 \(A_{m,k}:E_1(I_1)\to \mathbb R\), \(B_{n,j}:E_2(I_2)\to \mathbb R\). </p>
<p><div class="definition_thmwrapper " id="a0000000004">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.1</span>
  </div>
  <div class="definition_thmcontent">
  <p><em>For \(m,n\in \mathbb N\) define the sequences of operators \((L_m)_{m\geq 1}\) and \((K_n)_{n\geq 1}\) by <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">5</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">6</span>
</p>
</div> 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 \(I_2\). </em></p>

  </div>
</div> </p>
<p><div class="proposition_thmwrapper " id="a0000000005">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2.2</span>
  </div>
  <div class="proposition_thmcontent">
  <p>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. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>The proof follows immediately. </p>
<p>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 </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} \lim _{m\to \infty }\tfrac {(T_{m,j}L_m)(x)}{m^{\alpha _j}}= a_j(x) \end{equation}
  </div>
  <span class="equation_label">7</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_{n,j}K_n)(y)}{n^{\beta _j}}= b_j(y) \end{equation}
  </div>
  <span class="equation_label">8</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 \left\{ 0,1,\dots , \tfrac {s}{2}\right\} \right\} \! , \end{equation}
  </div>
  <span class="equation_label">9</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">10</span>
</p>
</div>
<p> where \(l\in \left\{ 0,1,2,\dots \tfrac {s}{2}\right\} \). </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">11</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><div class="definition_thmwrapper " id="a0000000007">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.3</span>
  </div>
  <div class="definition_thmcontent">
  <p><em>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 <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">12</span>
</p>
</div> is named the bivariate operator of \(LK\)-type. </em></p>

  </div>
</div> </p>
<p><div class="proposition_thmwrapper " id="a0000000008">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2.4</span>
  </div>
  <div class="proposition_thmcontent">
  <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>

  </div>
</div> <div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>The proof follows immediately. </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_{m,0}L_m)(x)=A_{m,0}(e_0)=1 \end{equation}
  </div>
  <span class="equation_label">13</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_{n,0}K_n)(y)=B_{n,0}(e_0)=1 \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> for any \(y\in I_2\cap J_2\), \(n\in \mathbb N\). </p>
<p>From <a href="#2.9" class="eqref">13</a>, <a href="#2.10" class="eqref">14</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">15</span>
</p>
</div>
<p> for any \(x\in I_1\cap J_1\), \(m\in \mathbb N\) and </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">16</span>
</p>
</div>
<p> for any \(y\in I_2\cap J_2\), \(n\in \mathbb N\). </p>
<p><div class="remark_thmwrapper " id="a0000000010">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.5</span>
  </div>
  <div class="remark_thmcontent">
  <p><em>From <a href="#2.9" class="eqref">13</a> and <a href="#2.10" class="eqref">14</a> it results that \(\alpha _0=\beta _0=0\).<span class="qed">â–¡</span></em></p>

  </div>
</div> </p>
<h1 id="a0000000011">3 Main results</h1>
<p>We recall the following theorem from [5]. </p>
<p><div class="theorem_thmwrapper " id="a0000000012">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>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 </p>
<div class="displaymath" id="3.1">
  \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}
</div>
<p> where </p>
<div class="displaymath" id="3.2">
  \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}
</div>
<p> \(k\in \{ 0,1,\dots ,n\} \), \(\mu \) is a bounded function with \(\lim \limits _{(x,y)\to (a,b)}\mu (x-a, y-b)=0\) and </p>
<div class="equation" id="3.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.3} \rho (x,y)=\sqrt{(x-a)^2+(y-b)^2}. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> Then for any \(\delta _1, \delta _2{\gt}0\), any \((x,y)\in V\) we have </p>
<div class="displaymath" id="3.4">
  \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}
</div>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000013">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f:I_1\times I_2\to \mathbb R\) be a bivariate function.<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="displaymath" id="3.5">
  \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}
</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="3.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.6} \tfrac {(T_{m, 2l}L_m)(x)}{m^{\alpha _{2l}}}\leq a_{2l}, \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<div class="equation" id="3.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.7} \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l}, \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p> 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 </p>
<div class="displaymath" id="3.8">
  \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}
</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="3.9">
  \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}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000014">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(m,n\in \mathbb N\). According to Taylor’s formula for the function \(f\) around \((x,y)\), we have </p>
<div class="displaymath" id="a0000000015">
  \[  f(t, \tau )=\sum ^s_{i=0}\tfrac {1}{i!}\left(\tfrac {\partial }{\partial t}(t-x)+\tfrac {\partial }{\partial \tau }(\tau -y)\right)^i f(x,y)+\rho ^s(t,\tau )\mu (t-x, \tau -y),  \]
</div>
<p> from where </p>
<div class="displaymath" id="3.10">
  \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}
</div>
<p> 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 (<a href="#2.7">11</a>), from (<a href="#3.10">27</a>) we have </p>
<div class="displaymath" id="a0000000016">
  \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*}
</div>
<p> 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 </p>
<div class="displaymath" id="a0000000017">
  \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*}
</div>
<p> from which </p>
<div class="displaymath" id="3.11">
  \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}
</div>
<p> where </p>
<div class="equation" id="3.12">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p> Then </p>
<div class="displaymath" id="a0000000018">
  \[  |(R_{m,m}f)(x,y)|\leq m^{s-\gamma _s}\sum ^{p_m}_{k=0} \sum ^{q_m}_{j=0}\varphi _{m,k}(x)\psi _{m,j}(y)\big|A_{m,m,k,j}\left(\rho ^s(\cdot , \ast )\mu _{xy}\right)\big|, \]
</div>
<p> from where </p>
<div class="displaymath" id="3.13">
  \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}
</div>
<p>According to the relation (<a href="#3.4">20</a>), 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 </p>
<div class="displaymath" id="a0000000019">
  \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*}
</div>
<p> 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, (<a href="#3.14">31</a>) results </p>
<div class="displaymath" id="3.14">
  \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}
</div>
<p> From (<a href="#3.13">30</a>) and (<a href="#3.14">31</a>), it results that </p>
<div class="displaymath" id="a0000000020">
  \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*}
</div>
<p> or </p>
<div class="displaymath" id="a0000000021">
  \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*}
</div>
<p> so </p>
<div class="displaymath" id="a0000000022">
  \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*}
</div>
<p> 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 (<a href="#2.5">9</a>) and from the inequalities above, we have </p>
<div class="displaymath" id="a0000000023">
  \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*}
</div>
<p> where \(l\in \bigg\{ 0,1,\dots , \tfrac {s}{2}\bigg\} \). </p>
<p>Considering \(\delta _1=\delta _2=\tfrac {1}{\sqrt{m^{\delta _s}}}\) , we have </p>
<div class="displaymath" id="3.15">
  \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}
</div>
<p> Taking (<a href="#2.3">7</a>), (<a href="#2.4">8</a>) into account and considering the fact that </p>
<div class="displaymath" id="a0000000024">
  \[  \lim _{m\to \infty }\omega _{total} \left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ; \tfrac {1}{\sqrt{m^{\delta _s}}}\, ,\tfrac {1}{\sqrt{m^{\delta _s}}}\right)= \omega _{total}\left(\tfrac {\partial ^s f}{\partial t^{s-i}\partial \tau ^i}\, ;0,0\right)=0,  \]
</div>
<p> \(i\in \{ 0,1,\dots ,s\} \), from (<a href="#3.15">34</a>) we have that </p>
<div class="equation" id="3.16">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.16} \lim _{m\to \infty }(R_{m,m}f)(x,y)=0. \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> From (<a href="#3.11">28</a>) and (<a href="#3.16">36</a>), (<a href="#3.6">23</a>) follows. </p>
<p>If in addition (<a href="#3.7">24</a>), (<a href="#3.8">25</a>) take place, then (<a href="#3.15">34</a>) becomes </p>
<div class="displaymath" id="3.17">
  \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}
</div>
<p> 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 (<a href="#3.5">22</a>) is uniform on \(K_1\times K_2\). From (<a href="#3.11">28</a>) and (<a href="#3.17">37</a>), (<a href="#3.8">25</a>) follows. <div class="proof_wrapper" id="a0000000025">
  <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="corollary_thmwrapper " id="a0000000026">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3.3</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(f:I_1\times I_2\to \mathbb R\) be a bivariate function.<br />If \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\) and \(f\) is continuous in \((x,y)\), then </p>
<div class="equation" id="3.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.18} \lim _{m\to \infty }\left(L^\ast _{m,m}f\right)(x,y)=f(x,y). \end{equation}
  </div>
  <span class="equation_label">38</span>
</p>
</div>
<p> 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 </p>
<div class="equation" id="3.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.19} \tfrac {(T_{m,2}L_m)(x)}{m^{\alpha _2}}\leq a_2, \end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<div class="equation" id="3.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.20} \tfrac {(T_{m,2}K_m)(y)}{m^{\beta _2}}\leq b_2, \end{equation}
  </div>
  <span class="equation_label">40</span>
</p>
</div>
<p> then the convergence given in \((\ref{3.18})\) is uniform on \(K_1\times K_2\) and </p>
<div class="equation" id="3.21">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">41</span>
</p>
</div>
<p> for any \((x,y)\in K_1\times K_2\), any \(m\in \mathbb N\), \(m\geq m(0)\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000027">
  <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 result from Theorem 3.2 for \(s=0\) and one verifies immediately that \(\alpha _0\! =\! \beta _0\! =\! \gamma _0=0\), \(a_0\! =\! b_0\! =\! 1\), \(\delta _0\! =\! -\max \bigg\{ \beta _2\! -\! 2, \alpha _2\! -\! 2, \tfrac {1}{2}\, (\alpha _2\! +\! \beta _2\! -\! 4)\bigg\} \). </p>
<p>In the following, in addition we suppose that </p>
<div class="equation" id="3.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.22} \alpha _{s+2}<\alpha _s+2,\;  \beta _{s+2}<\beta _s+2 \end{equation}
  </div>
  <span class="equation_label">42</span>
</p>
</div>
<p> and for any \(f\in E(I_1\times I_2)\) we have </p>
<div class="equation" id="3.23">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.23} A_{m,n,k,j}(f_x)=B_{n,j}(f_x), \end{equation}
  </div>
  <span class="equation_label">43</span>
</p>
</div>
<div class="equation" id="3.24">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.24} A_{m,n,k,j}(f^y)=A_{m,k}(f^y), \end{equation}
  </div>
  <span class="equation_label">44</span>
</p>
</div>
<div class="equation" id="3.25">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">45</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>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 \(I=I_2\), \(J=J_2\), \(\eta _j=\beta _j\), \(k_j=b_j\), \(j\in \{ s, s+2\} \). </p>
<p><div class="theorem_thmwrapper " id="a0000000028">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>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 </p>
<div class="equation" id="3.26">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">46</span>
</p>
</div>
<p> 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 </p>
<div class="equation" id="3.27">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.27} \tfrac {(T_{m,j}P_m)(x)}{m^{\eta _j}}\leq k_j \end{equation}
  </div>
  <span class="equation_label">47</span>
</p>
</div>
<p> where \(j\in \{ s, s+2\} \). Then the convergence given in \((\ref{3.26})\) is uniform on \(M\) and </p>
<div class="displaymath" id="3.28">
  \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}
</div>
<p> for any \(x\in M\) and \(m\geq m(s)\). </p>

  </div>
</div> </p>
<p>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. </p>
<p><div class="lemma_thmwrapper " id="a0000000029">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.5</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="3.29">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">48</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> </p>
<p><div class="proof_wrapper" id="a0000000030">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> We have </p>
<div class="displaymath" id="a0000000031">
  \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*}
</div>
<p> and taking (<a href="#3.24">44</a>), (<a href="#3.25">45</a>) into account, we obtain </p>
<div class="displaymath" id="a0000000032">
  \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*}
</div>
<p> From (<a href="#2.1">5</a>), (<a href="#2.2">6</a>), (<a href="#2.11">15</a>) and (<a href="#2.12">16</a>), the relation (<a href="#3.29">48</a>) is obtained. <div class="proof_wrapper" id="a0000000033">
  <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="a0000000034">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f:I_1\times I_2\to \mathbb R\) be a bivariate function.<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="displaymath" id="3.30">
  \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}
</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 \(x\in K_1\), \(y\in K_2\) we have </p>
<div class="equation" id="3.31">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.31} \tfrac {\left(T_{m,2l}L_m\right)(x)}{m^{\alpha _{2l}}} \leq a_{2l}, \end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<div class="equation" id="3.32">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.32} \tfrac {(T_{m,2l}K_m)(y)}{m^{\beta _{2l}}}\leq b_{2l} \end{equation}
  </div>
  <span class="equation_label">51</span>
</p>
</div>
<p> 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 </p>
<div class="displaymath" id="3.33">
  \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}
</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 \((\ref{3.9})\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000035">
  <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="#2.5">9</a>), 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 (<a href="#3.26">46</a>) 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 (<a href="#3.5">22</a>) relation from Theorem 3.2 for the function \(f\) and then we obtain the (<a href="#3.30">49</a>) relation. If we note by \(S\) the left member of (<a href="#3.33">52</a>) relation, we can write </p>
<div class="displaymath" id="a0000000036">
  \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*}
</div>
<div class="displaymath" id="a0000000037">
  \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*}
</div>
<p> and taking Theorem 3.4 and relation (<a href="#3.8">25</a>) into account we obtain the first inequality from (<a href="#3.33">52</a>). From (<a href="#3.33">52</a>) the uniform convergence for (<a href="#3.30">49</a>) results. <div class="proof_wrapper" id="a0000000038">
  <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="corollary_thmwrapper " id="a0000000039">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3.7</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(f:I_1\times I_2\to \mathbb R\) be a bivariate function. </p>
<p>If \((x,y)\in (I_1\cap J_1)\times (I_2\cap J_2)\) and \(f\) is continuous in \((x,y)\), then </p>
<div class="equation" id="3.34">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.34} \lim _{m\to \infty }\left(UL^\ast _{m,m} f\right)(x,y)=f(x,y). \end{equation}
  </div>
  <span class="equation_label">53</span>
</p>
</div>
<p>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 </p>
<div class="equation" id="3.35">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.35} \tfrac {\left(T_{m,2}L_m\right)(x)}{m^{\alpha _2}} \leq a_2 \end{equation}
  </div>
  <span class="equation_label">54</span>
</p>
</div>
<p> and </p>
<div class="equation" id="3.36">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.36} \tfrac {(T_{m,2}K_m)(y)}{m^{\beta _2}}\leq b_2. \end{equation}
  </div>
  <span class="equation_label">55</span>
</p>
</div>
<p> Then the convergence given in \((\ref{3.34})\) is uniform on \(K_1\times K_2\) and </p>
<div class="displaymath" id="3.37">
  \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}
</div>
<p> for any \((x,y)\in K_1\times K_2\) and any \(m\in \mathbb N\), \(m\geq m(0)\), where </p>
<div class="displaymath" id="a0000000040">
  \[  \delta _0=-\max \bigg\{ \beta _2-2, \alpha _2-2, \tfrac {1}{2}\, (\alpha _2+\beta _2-4)\bigg\} .  \]
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000041">
  <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 3.6 for \(s=0\). <div class="proof_wrapper" id="a0000000042">
  <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>Because every application is a simple substitute in the results of this section, we won’t replace anything. </p>
<p><div class="ap_thmwrapper " id="a0000000043">
  <div class="ap_thmheading">
    <span class="ap_thmcaption">
    Application
    </span>
    <span class="ap_thmlabel">3.8</span>
  </div>
  <div class="ap_thmcontent">
  <p><em>If \(I_1=J_1=[0,1]\), \(I_2=J_2=[0,\infty )\), \(E_1(I_1)=C([0,1])\), \(E_2(I_2)\! =\! C_2([0,\infty ))\), \(p_m\! =\! m\), \(q_n\! =\! \infty \), \(\varphi _{m,k}(x)\! =\! p_{m,k}(x)\! =\! \tbinom {m}{k}x^k(1-x)^{m-k}\),<br />\(\psi _{n,j}(y)={\rm e}^{-ny}\tfrac {(ny)^j}{j!}\) , \(A_{m,k}(f^y)=f\left(\tfrac {k}{m}\, , y\right)\), \(B_{n,j}(f_x)=f\left(x, \tfrac {j}{n}\right)\),<br />\(A_{m,n,j,k}(f)=f\left(\tfrac {k}{m}\, ,\tfrac {j}{n}\right)\) for any \((x,y)\in [0,1]\times [0,\infty )\), \(m,n\in \mathbb N\),<br />\(k\in \{ 0,1,\dots , m\} \), \(j\in \mathbb N_0\) 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 <div class="equation" id="3.38">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">57</span>
</p>
</div> <div class="equation" id="3.39">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">58</span>
</p>
</div> 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 \(m(0)=1\) <em>(see [3])</em>. </em></p>

  </div>
</div> </p>
<p><div class="ap_thmwrapper " id="a0000000044">
  <div class="ap_thmheading">
    <span class="ap_thmcaption">
    Application
    </span>
    <span class="ap_thmlabel">3.9</span>
  </div>
  <div class="ap_thmcontent">
  <p><em>If \(I_1=J_1=[0,\infty )\), \(I_2\! =\! J_2\! =\! [0,1]\), \(E_1(I_1)\! =\! C_2([0,\infty ))\),<br />\(E_2(I_2)\! =\! L_1([0,1])\), \(p_m\! =\! \infty \), \(q_n=n\), \(\varphi _{m,k}(x)\! =\! (1+x)^{-m} \tbinom {m\! +\! k\! -\! 1}{k}\big(\tfrac {x}{1\! +\! x}\big)^k\), \(\psi _{n,j}(y)\! =\! \tbinom {n}{j}y^j(1-y)^{n\! -\! j}\), <div class="displaymath" id="a0000000045">
  \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*}
</div> 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 <div class="displaymath" id="3.40">
  \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}
</div> <div class="equation" id="3.41">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">60</span>
</p>
</div> 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 \(m(0)=3\) <em>(see [2] and [3])</em>. </em></p>

  </div>
</div> </p>
<p><div class="ap_thmwrapper " id="a0000000046">
  <div class="ap_thmheading">
    <span class="ap_thmcaption">
    Application
    </span>
    <span class="ap_thmlabel">3.10</span>
  </div>
  <div class="ap_thmcontent">
  <p><em>If \(I_1=I_2=J_1=J_2=[0,1]\), \(E_1(I_1)=C([0,1])\), \(E_2(I_2)=L_1([0,1])\), \(p_m=\infty \), \(q_n=n\), \(\varphi _{m,k}(x)=\tbinom {m+k}{k}(1-x)^{m+1}x^k\), \(\psi _{n,j}(y)=p_{n,j}(y)\), <div class="displaymath" id="a0000000047">
  \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*}
</div> 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 <div class="displaymath" id="3.42">
  \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}
</div> <div class="equation" id="3.43">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">62</span>
</p>
</div> 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 \(m(0)=3\) <em>(see [2] and [3])</em>. </em></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
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  <dd><p><i class="sc">Badea, C.</i> and <i class="sc">Cottin, C.</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), pp.&#160;51–67, 1990. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no1-art9"> <i class="sc">Pop, O. T.</i>, <em>The generalization of Voronovskaja’s theorem for a class of linear and positive operators</em>, Rev. Anal. Num. Théor. Approx., <b class="bfseries">34</b>, No. 1, pp.&#160;79–91, 2005. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
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</dd>
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  <dd><p><i class="sc">Pop, O. T.</i>, <em>About operator of Bleimann, Butzer and Hahn</em>, Anal. Univ. Timişoara, <b class="bfseries">XLIII</b>, Fasc. 1, pp.&#160;117–127, 2005. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">Pop, O. T.</i>, <em>The generalization of Voronovskaja’s theorem for a class of bivariate operators</em>, anal. Univ. Oradea, Fasc. Matematica, Tom <b class="bfseries">XV</b>, pp.&#160;155–169, 2008. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no2-art6"> <i class="sc">Pop, O. T.</i>, <em>About a general property for a class of linear positive operators and applications</em>, Rev. Anal. Num. Théor. Approx., <b class="bfseries">34</b>, No. 2, pp.175–180, 2005. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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</dd>
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  <dd><p><i class="sc">Pop, O. T.</i>, <em>Voronovskaja-type theorems and approximation theorems for a class of GBS operators</em>, Fasc. Math., <b class="bf">42</b>, pp.&#160;91–108, 2009. </p>
</dd>
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  <dd><p><i class="sc">Pop, O. T.</i>, <em>Voronovskaja-type theorem for certain GBS operators</em>, Glasnik Matematički, <b class="bfseries">43</b> (63), pp.&#160;179–194, 2008. </p>
</dd>
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  <dd><p><i class="sc">Stancu, D. D., Coman, Gh., Agratini, O.</i> and <i class="sc">Trîmbiţaş, R.</i>, <em>Analiză numerică şi teoria aproximării</em>, I, Presa Universitară Clujeană, Cluj-Napoca, 2001 (in Romanian). </p>
</dd>
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</dd>
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  <dd><p><i class="sc">Voronovskaja, E.</i>, <em>Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein</em>, C. R. Acad. Sci. URSS, pp.&#160;79–85, 1932. </p>
</dd>
</dl>


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