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<title>The Kantorovich form of some extensions for Szász-Mirakjan operators: The Kantorovich form of some extensions for Szász-Mirakjan operators</title>
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<h1>The Kantorovich form of some extensions for Szász-Mirakjan operators</h1>
<p class="authors">
<span class="author">Dan Bărbosu\(^\ast \), Ovidiu T. Pop\(^\S \) Dan Miclăuş\(^\ast \)</span>
</p>
<p class="date">September 23, 2009.</p>
</div>
<p>\(^\ast \)Department of Mathematics and Computer Science, Faculty of Sciences, North University of Baia Mare, Victoriei 76, 430122 Baia Mare, Romania, e-mail: <span class="tt">{barbosudan, danmiclausrz}@yahoo.com</span> </p>
<p>\(^\S \)National College “Mihai Eminescu”, 5 Mihai Eminescu Street, 440014 Satu Mare, Romania, e-mail: <span class="tt">ovidiutiberiu@yahoo.com</span> </p>

<div class="abstract"><p> Recently, C. Mortici defined a class of linear and positive operators depending on a certain function \(\varphi \). These operators generalize the well known Szász-Mirakjan operators. A convergence theorem for the defined sequence by the mentioned operators was given. Other interesting approximation properties of these generalized Szász-Mirakjan operators and also their bivariate form were obtained by D. Bărbosu, O. T. Pop and D. Miclăuş. In the present paper we are dealing with the Kantorovich form of the generalized Szász-Mirakjan operators. We construct the Kantorovich associated operators and then we establish a convergence theorem for the defined operators. The degree of approximation is expressed in terms of the modulus of continuity. Next, we construct the bivariate and respectively the GBS corresponding operators and we establish some of their approximation properties. </p>
<p><b class="bf">MSC.</b> 41A10, 41A25, 41A36. </p>
<p><b class="bf">Keywords.</b> Szász-Mirakjan operators, Kantorovich operators, Bohman-Korovkin theorem, modulus of continuity, Shisha-Mond theorem, degree of approximation, parametric extension, Korovkin theorem for the bivariate case, bivariate modulus of continuity, Bögel continuity, Korovkin theorem for B-continuous functions, mixed modulus of smoothness, Shisha-Mond theorem for the B-continuous functions. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\mathbb N\) be the set of positive integers and \(\mathbb N_{0}=\mathbb N\cup \{ 0\} \). </p>
<p>Considering an analytic function \(\varphi :\mathbb {R}\to ]0,+\infty [\), C. Mortici [19] defined the operators </p>
<div class="displaymath" id="a0000000003">
  \[ \varphi S_{n}:C^{2}([0,+\infty [)\to C^{\infty }([0,+\infty [),  \]
</div>
<p> given by </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} (\varphi S_{n}f)(x)=\tfrac {1}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}f\left(\tfrac {k}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p> for any \(x \in [0,+\infty [\) and any \(n\in \mathbb N\). </p>
<p>The operators <a href="#1.1" class="eqref">1.1</a> are called the \(\varphi \)-Szász-Mirakjan operators, because in the case when \(\varphi (y)={\rm e}^{y}\), they become the classical Mirakjan-Favard-Szász operators [2], [14], [18], [23], [25]. Some nice and interesting approximation properties of operators <a href="#1.1" class="eqref">1.1</a> were obtained by C. Mortici [19] and by D. Bărbosu, O. T. Pop and D. Miclăuş [8], [20]. </p>
<p><div class="remark_thmwrapper " id="a0000000004">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.1</span>
  </div>
  <div class="remark_thmcontent">
  <p>Similar generalization of this type are the operators defined and studied by Jakimovski and Leviatan [15] or the operators defined by Baskakov in 1957 (see, e.g., the book [2], subsection 5.3.11, p. 344, where they are attributed to Mastroianni). </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000005">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>The classical Mirakjan-Favard-Szász operators \(S_n:\)<br />\(C_2([0,+\infty [)\to C([0,+\infty [)\) are defined by </p>
<div class="displaymath" id="a0000000006">
  \[ (S_nf)(x)=e^{-nx}\sum \limits _{k=0}^{\infty }\tfrac {(nx)^k}{k!}f\left(\tfrac {k}{n}\right), \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000007">
  \[ C_2([0,+\infty [):=\left\{ f\in C([0,+\infty [):\lim \limits _{x\to \infty }\tfrac {f(x)}{1+x^2}\  \mbox{exists\  and\  is\  finite}\right\} . \]
</div>
<p> In what follows, we shall use the classical definition of Mirakjan-Favard-Szász operators, i.e. \(f\in C_2([0,+\infty [)\). </p>

  </div>
</div> </p>
<p>The main goal of the present paper is to construct the Kantorovich type operators, associated to the \(\varphi \)-Szász-Mirakjan operators <a href="#1.1" class="eqref">1.1</a>. </p>
<p>Using the method of parametric extensions [7], [12], the bivariate \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Katorovich operators are constructed and some of their approximation properties are established. The last section is devoted to the construction of the associated GBS \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operators and to study some of their approximation properties. </p>
<h1 id="a0000000008">2 \(\varphi \)-Szász-Mirakjan-Katorovich operators</h1>
<p>Let \(\varphi :\mathbb R\to ]0,+\infty [\) be an analytic function. Following the idea of L. V. Kantorovich [16] we define the operators </p>
<div class="displaymath" id="a0000000009">
  \[ \varphi K_{n}:C_{2}([0,+\infty [)\to C([0,+\infty [), \]
</div>
<p> given by </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} (\varphi K_{n}f)(x)=\tfrac {n}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}f(t){\rm d}t, \end{equation}
  </div>
  <span class="equation_label">2.2</span>
</p>
</div>
<p> for any \(x\in [0,+\infty [\) and \(n\in \mathbb N\). </p>
<p>The operators <a href="#2.1" class="eqref">2.2</a> will be called the \(\varphi \)-Szász-Mirakjan-Kantorovich operators, because in the case when \(\varphi (y)={\rm e}^{y}\), they reduce to the classical Szász-Mirakajan-Kantorovich operators [1], [17]. </p>
<p><div class="remark_thmwrapper " id="a0000000010">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.1</span>
  </div>
  <div class="remark_thmcontent">
  <p>The operators <a href="#2.1" class="eqref">2.2</a> are linear and positive.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>In order to obtain the convergence of the sequence \((\varphi K_{n})_{n\in \mathbb N}\) we need the following: </p>
<p><div class="lemma_thmwrapper " id="a0000000011">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(e_{j}(x)=x^{j}\), \(j=0,1,2\) be the test functions. The \(\varphi \)-Szász-Mirakjan-Kantorovich operators satisfy the following relations: </p>
<ul class="itemize">
  <li><p>\((\varphi K_{n}e_{0})(x)=1,\) </p>
</li>
  <li><p>\((\varphi K_{n}e_{1})(x)=\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{2n},\) </p>
</li>
  <li><p>\((\varphi K_{n}e_{2})(x)=\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}x^{2}+\tfrac {2}{n}\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{3n^{2}}.\) </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Taking that the function \(\varphi \) is analytic into account, it follows </p>
<div class="displaymath" id="a0000000013">
  \[ \sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}y^{k}=\varphi (y) \]
</div>
<p> and next, by differentiation </p>
<div class="displaymath" id="a0000000014">
  \[ \sum \limits _{k=1}^{\infty }\tfrac {\varphi ^{(k)}(0)}{(k-1)!}y^{k-1}=\varphi ^{(1)}(y), \  \  \  \  \sum \limits _{k=2}^{\infty }\tfrac {\varphi ^{(k)}(0)}{(k-2)!}y^{k-2}=\varphi ^{(2)}(y). \]
</div>
<p> For the test functions \(e_{0},e_{1},e_{2}\), the following identities </p>
<div class="displaymath" id="a0000000015">
  \[ \int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{0}(t){\rm d}t=\tfrac {1}{n}, \  \  \  \  \int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{1}(t){\rm d}t=\tfrac {2k+1}{2n^{2}}, \  \  \  \  \  \int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{2}(t){\rm d}t=\tfrac {3k^{2}+3k+1}{3n^{3}} \]
</div>
<p> hold. </p>
<p>Recall that, the images of test functions by the operators \(\varphi \)-Szász-Mirakjan [19] are </p>
<div class="displaymath" id="2.2">
  \begin{align} \label{2.2} &  \nonumber (\varphi S_{n}e_{0})(x)=1,\\ &  (\varphi S_{n}e_{1})(x)=\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x,\\ \nonumber &  (\varphi S_{n}e_{2})(x)=\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}x^{2}+\tfrac {1}{n}\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x,\\ \nonumber \end{align}
</div>
<p> then </p>
<div class="displaymath" id="a0000000016">
  \begin{align*}  (\varphi K_{n}e_{0})(x)& =\tfrac {n}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{0}(t){\rm d}t\\ & =\tfrac {n}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\tfrac {1}{n}\\ & =\tfrac {1}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\\ & =(\varphi S_{n}e_{0})(x)=1. \end{align*}
</div>
<div class="displaymath" id="a0000000017">
  \begin{align*}  (\varphi K_{n}e_{1})(x)& =\tfrac {n}{\varphi (nx)}\! \! \sum \limits _{k=0}^{\infty }\! \tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\! \! \int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{1}(t){\rm d}t\! \\ & =\tfrac {n}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\tfrac {2k+1}{2n^{2}}\\ & =\tfrac {1}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\tfrac {k}{n}+\tfrac {1}{2n}\tfrac {1}{\varphi (nx)} \sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\\ & =(\varphi S_{n}e_{1})(x)+\tfrac {1}{2n}(\varphi S_{n}e_{0})(x) \\ & =\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{2n}. \end{align*}
</div>
<div class="displaymath" id="a0000000018">
  \begin{align*}  (\varphi K_{n}e_{2})(x)& =\tfrac {n}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\int _{\tfrac {k}{n}}^{\tfrac {k+1}{n}}e_{2}(t){\rm d}t\\ & =\tfrac {1}{n^{2}}\tfrac {1}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\left(k^{2}+k+\tfrac {1}{3}\right)\\ & =\tfrac {1}{n^{2}}\left(\tfrac {n^{2}}{\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\tfrac {k^2}{n^2}\right.\\ & \qquad \quad \left.+\tfrac {n}{\varphi (nx)} \sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\tfrac {k}{n}+\tfrac {1}{3\varphi (nx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi ^{(k)}(0)}{k!}(nx)^{k}\right)\\ & =(\varphi S_{n}e_{2})(x)+\tfrac {(\varphi S_{n}e_{1})(x)}{n}+\tfrac {(\varphi S_{n}e_{0})(x)}{3n^2}\\ & =\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}x^{2}+\tfrac {2}{n}\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{3n^{2}}. \end{align*}
</div>
<p>Applying Lemma 2.2. we shall prove the following: <div class="theorem_thmwrapper " id="a0000000019">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\varphi :\mathbb R\to ]0,+\infty [\) be an analytic function having the properties </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} \lim \limits _{y\to \infty }\tfrac {\varphi ^{(1)}(y)}{\varphi (y)}=1,\  \   \lim \limits _{y\to \infty }\tfrac {\varphi ^{(2)}(y)}{\varphi (y)}=1. \end{equation}
  </div>
  <span class="equation_label">2.4</span>
</p>
</div>
<p> Then, for any function \(f\in C_{2}([0,+\infty [)\) and any \(x\in [a,b]\), it holds </p>
<div class="displaymath" id="a0000000020">
  \[ \lim \limits _{n\to \infty }(\varphi K_{n}f)(x)=f(x) \]
</div>
<p> uniformly on \([a,b]\), where \([a,b]\subset [0,+\infty [\) is a compact interval. </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>Applying the Lemma 2.2. and the hypothesis <a href="#2.3" class="eqref">2.4</a>, it follows </p>
<div class="displaymath" id="a0000000022">
  \[ \lim \limits _{n\to \infty }(\varphi K_{n}e_{1})(x)=\lim \limits _{n\to \infty }\left(\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{2n}\right)=x \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000023">
  \begin{align*}  \lim \limits _{n\to \infty }(\varphi K_{n}e_{2})(x)=\lim \limits _{n\to \infty }\left(\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}x^2+\tfrac {2}{n}\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}x+\tfrac {1}{3n^{2}}\right)=x^2 \end{align*}
</div>
<p> uniformly on any compact interval \([a,b]\subset [0,+\infty [.\) </p>
<p>Next, from the well known Bohman-Korovkin Theorem one arrives to the desired result. In order to obtain the degree of approximation of \(f\in C_{2}([0,+\infty [)\), by means of the \(\varphi \)-Szász-Mirakjan-Kantorovich operators, let us to recalling some known results, concerning the modulus of continuity. </p>
<p>Let \(I\subset \mathbb {R}\) be an interval, \(C(I)\) be the set of real-valued functions continuous on \(I\), \(B(I)\) be the set of real-valued functions bounded on \(I\) and \(C_{B}(I)\) be the set of real-valued functions continuous, bounded on \(I\). </p>
<p><div class="definition_thmwrapper " id="a0000000024">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.4</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(f\in C_{B}(I)\) be given. The function \(\omega _{1}:[0,+\infty [\to \mathbb {R}\) defined by </p>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} \omega _{1}(f;\delta ):=\sup \{ |f(x+h)-f(x)|\; :\; x,\; x+h\in I, \  0<h\leq \delta \}  \end{equation}
  </div>
  <span class="equation_label">2.5</span>
</p>
</div>
<p> is called the modulus of continuity (the first modulus of smoothness) of the function \(f\). </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000025">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.5</span>
  </div>
  <div class="remark_thmcontent">
  <p><em>Its properties can be found in the monograph <em>[1]</em>.<span class="qed">â–¡</span></em></p>

  </div>
</div> In 1968, O. Shisha and B. Mond [21] established the following: <div class="theorem_thmwrapper " id="a0000000026">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>  <em>[1]</em> Let \(L:C(I)\to B(I)\) be a linear positive operator and let the function \(\varphi \) be defined by </p>
<div class="displaymath" id="a0000000027">
  \[ \varphi _{x}(t)=|t-x|, \  (x,t)\in I\times I. \]
</div>
<p> If \(f\in C_{B}(I)\), then for any \(x\in I\) and \(\delta {\gt}0\) the following </p>
<div class="displaymath" id="2.5">
  \begin{align} \label{2.5} |(Lf)(x)-f(x)| &  \leq |f(x)||(Le_{0})(x)-1|\\ &  \  \  \  +\left((Le_{0})(x)+\delta ^{-1}\sqrt{(Le_{0})(x)(L\varphi _{x}^{2})(x)}\right)\omega _{1}(f;\delta )\nonumber \end{align}
</div>
<p> holds. </p>

  </div>
</div> For obtaining the degree of approximation of \(f\in C_2([0,+\infty [)\), on any compact interval \([a,b]\subset [0,+\infty [\), by means of the \(\varphi \)-Szász-Mirakjan-Kantorovich operators we need the following: </p>
<p><div class="lemma_thmwrapper " id="a0000000028">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.7</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let the function \(\varphi _{x}\) be defined by \(\varphi _{x}=|t-x|\), \((x,t)\in [a,b]\times [a,b].\) The operators <a href="#2.1" class="eqref">2.2</a> verify the relation </p>
<div class="displaymath" id="2.6">
  \begin{align} \label{2.6} (\varphi K_{n}\varphi _{x}^2)(x) =\left(\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}-2\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}+1\right)x^2 +\tfrac {1}{n}\left(2\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}-1\right)x+\tfrac {1}{3n^{2}}. \end{align}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000029">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Because the operators \(\varphi \)-Szász-Mirakjan-Kantorovich are linear, then taking the definition of \(\varphi _{x}\) into account, we get </p>
<div class="displaymath" id="a0000000030">
  \[ (\varphi K_{n}\varphi _{x})(x)=(\varphi K_{n}e_{2})(x)-2x(\varphi K_{n}e_{1})(x)+x^{2}(\varphi K_{n}e_{0})(x). \]
</div>
<p> Next, one applies Lemma 2.2. In the following, we suppose that the analytic function \(\varphi :\mathbb R\to ]0,+\infty [\) satisfy the conditions <a href="#2.3" class="eqref">2.4</a> and taking these conditions into account, it results </p>
<div class="displaymath" id="a0000000031">
  \[ \lim \limits _{n\to \infty }\left(\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}-2\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}+1\right)=0. \]
</div>
<p> Then we suppose that, there exists \(0{\lt}\gamma \leq 1\), so that </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} \lim \limits _{n\to \infty }n^{\gamma }\left(\tfrac {\varphi ^{(2)}(nx)}{\varphi (nx)}-2\tfrac {\varphi ^{(1)}(nx)}{\varphi (nx)}+1\right)=\beta _{2}(x), \end{equation}
  </div>
  <span class="equation_label">2.8</span>
</p>
</div>
<p> for any \(x\in [0,+\infty [\) and where \(\beta _{2}\) is a function, \(\beta _{2}:[0,+\infty [\to \mathbb R\). </p>
<p><div class="theorem_thmwrapper " id="a0000000032">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any function \(f\in C_{2}([0,+\infty [)\), any compact interval \([a,b]\subset [0,+\infty [\), any \(x\in [a,b]\) and any \(\delta {\gt}0\), the \(\varphi \)-Szász-Mirakjan-Kantorovich operators verify the inequality </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} |(\varphi K_{n}f)(x)-f(x)|\leq K\omega _{1}\left(f;\tfrac {b-a}{\sqrt{n^{\gamma }}}\right), \end{equation}
  </div>
  <span class="equation_label">2.9</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000033">
  \[ m_{2}([a,b]):=\sup \limits _{x\in [a,b]}|\beta _{2}(x)| \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000034">
  \[ K=1+\tfrac {1}{b-a}\sqrt{(m_{2}([a,b])+1)b^{2}+2b+\tfrac {1}{3}}. \]
</div>

  </div>
</div> <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>The relation <a href="#2.8" class="eqref">2.9</a> yields from <a href="#2.5" class="eqref">2.6</a>, if we choose \(\delta =\tfrac {b-a}{\sqrt{n^{\gamma }}}\) and if we take the definition of limit and relation <a href="#2.7" class="eqref">2.8</a> into account. </p>
<h1 id="a0000000036">3 The bivariate \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operators</h1>
<p>Suppose that \(\varphi _{1},\varphi _{2}:\mathbb {R}\to ]0,+\infty [\) are analytic functions. The operators </p>
<div class="displaymath" id="a0000000037">
  \[ \varphi _{1}K_{m}^{x}:C_{2,2}([0,+\infty [\times [0,+\infty [)\to C([0,+\infty [\times [0,+\infty [), \]
</div>
<div class="displaymath" id="a0000000038">
  \[ \varphi _{1}K_{n}^{y}:C_{2,2}([0,+\infty [\times [0,+\infty [)\to C([0,+\infty [\times [0,+\infty [), \]
</div>
<p> given by </p>
<div class="displaymath" id="3.1">
  \begin{align} \label{3.1} (\varphi _{1}K_{m}^{x}f)(x,y)& =\tfrac {m}{\varphi _{1}(mx)}\sum \limits _{k=0}^{\infty }\tfrac {\varphi _{1}^{(k)}(0)}{k!}(mx)^{k}\int _{\tfrac {k}{m}}^{\tfrac {k+1}{m}}f(t,y){\rm d}t, \\ \label{3.2} (\varphi _{1}K_{n}^{y}f)(x,y)& =\tfrac {n}{\varphi _{2}(ny)}\sum \limits _{j=0}^{\infty }\tfrac {\varphi _{2}^{(j)}(0)}{j!}(ny)^{j}\int _{\tfrac {j}{n}}^{\tfrac {j+1}{n}}f(x,s){\rm d}s \end{align}
</div>
<p> are called the parametric extensions of the \(\varphi \)-Szász-Mirakjan-Kantorovich operators; (for the notion of “parametric extensions" see [7], [12]). </p>
<p>It is immediately the result contained in the following: </p>
<p><div class="lemma_thmwrapper " id="a0000000039">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p>The parametric extension of the \(\varphi \)-Szász-Mirakjan-Kantorovich operators defined at <a href="#3.1" class="eqref">3.10</a> and <a href="#3.2" class="eqref">3.11</a> are linear and positive. They commute on \(C_{2,2}([0,+\infty [\times [0,+\infty [)\) and their product are the bivariate \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operators </p>
<div class="displaymath" id="a0000000040">
  \[  \varphi _{1}\varphi _{2}K_{m,n}:C_{2,2}([0,+\infty [\times [0,+\infty [)\to C([0,+\infty [\times [0,+\infty [)  \]
</div>
<p> defined by </p>
<div class="displaymath" id="3.3">
  \begin{align} \label{3.3} & (\varphi _{1}\varphi _{2}K_{m,n}f)(x,y)= \\ & =\tfrac {m}{\varphi _{1}(mx)}\tfrac {n}{\varphi _{2}(ny)}\sum \limits _{k=0}^{\infty }\sum \limits _{j=0}^{\infty }\tfrac {\varphi _{1}^{(k)}(0)}{k!}\tfrac {\varphi _{2}^{(j)}(0)}{j!} (mx)^{k}(ny)^{j}\! \int _{\tfrac {k}{m}}^{\tfrac {k+1}{m}}\! \int _{\tfrac {j}{n}}^{\tfrac {j+1}{n}}\! f(t,s){\rm d}t{\rm d}s\nonumber , \end{align}
</div>
<p> for any \(x,y\in [0,+\infty [\) and \(m,n\in \mathbb N\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000041">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(e_{ij}(x,y)=x^{i}y^{j}\), \(i,j\in \mathbb N_{0},\) \(i+j\leq 2\) be the test functions. </p>
<p>The operators <a href="#3.3" class="eqref">3.12</a> verify the following identities: </p>
<ul class="itemize">
  <li><p>\((\varphi _{1}\varphi _{2}K_{m,n}e_{0,0})(x,y)=1,\) </p>
</li>
  <li><p>\((\varphi _{1}\varphi _{2}K_{m,n}e_{1,0})(x,y)=\tfrac {\varphi _{1}^{(1)}(mx)}{\varphi _{1}(mx)}x+\tfrac {1}{2m},\) </p>
</li>
  <li><p>\((\varphi _{1}\varphi _{2}K_{m,n}e_{0,1})(x,y)=\tfrac {\varphi _{2}^{(1)}(ny)}{\varphi _{2}(ny)}y+\tfrac {1}{2n},\) </p>
</li>
  <li><p>\((\varphi _{1}\varphi _{2}K_{m,n}e_{2,0})(x,y)=\tfrac {\varphi _{1}^{(2)}(mx)}{\varphi _{1}(mx)}x^{2}+\tfrac {2}{m}\tfrac {\varphi _{1}^{(1)}(mx)}{\varphi _{1}(mx)}x+\tfrac {1}{3m^2},\) </p>
</li>
  <li><p>\((\varphi _{1}\varphi _{2}K_{m,n}e_{0,2})(x,y)=\tfrac {\varphi _{2}^{(2)}(ny)}{\varphi _{2}(ny)}y^{2}+\tfrac {2}{n}\tfrac {\varphi _{2}^{(1)}(ny)}{\varphi _{2}(ny)}y +\tfrac {1}{3n^2}.\) </p>
</li>
</ul>

  </div>
</div> <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>One applies relation <a href="#3.3" class="eqref">3.12</a> and takes Lemma 2.2. into account. </p>
<p><div class="lemma_thmwrapper " id="a0000000043">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.3</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(\varphi _{x},\varphi _{y}:[0,+\infty [\times [0,+\infty [\to \mathbb {R}\) be defined by </p>
<div class="displaymath" id="a0000000044">
  \[ \varphi _{x}(s,t)=|s-x|, \  \  \varphi _{y}(s,t)=|t-y|. \]
</div>
<p> The operators <a href="#3.3" class="eqref">3.12</a> satisfy </p>
<div class="displaymath" id="3.4">
  \begin{align} \label{3.4} & (\varphi _{1}\varphi _{2}K_{m,n}\varphi _{x}^{2})(x,y) =\\ & =\left(\tfrac {\varphi _{1}^{(2)}(mx)}{\varphi _{1}(mx)}-2\tfrac {\varphi _{1}^{(1)}(mx)}{\varphi _{1}(mx)}+1\right)x^{2} +\tfrac {1}{m}\left(2\tfrac {\varphi _{1}^{(1)}(mx)}{\varphi _{1}(mx)}-1\right)x+\tfrac {1}{3m^2} \nonumber \end{align}
</div>
<p> and </p>
<div class="displaymath" id="3.5">
  \begin{align} \label{3.5} & (\varphi _{1}\varphi _{2}K_{m,n}\varphi _{y}^{2})(x,y) =\\ & =\left(\tfrac {\varphi _{2}^{(2)}(ny)}{\varphi _{2}(ny)}-2\tfrac {\varphi _{2}^{(1)}(ny)}{\varphi _{2}(ny)}+1\right)y^{2} +\tfrac {1}{n}\left(2\tfrac {\varphi _{2}^{(1)}(ny)}{\varphi _{2}(ny)}-1\right)y+\tfrac {1}{3n^2}\nonumber . \end{align}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000045">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Taking the linearity of operator <a href="#3.3" class="eqref">3.12</a> and the definitions of the functions \(\varphi _{x},\  \varphi _{y}\) into account, one obtains </p>
<div class="displaymath" id="a0000000046">
  \begin{align*}  (\varphi _{1}\varphi _{2}K_{m,n}\varphi _{x}^{2})(x,y) &  =(\varphi _{1}\varphi _{2}K_{m,n}e_{2,0})(x,y)-2x(\varphi _{1}\varphi _{2}K_{m,n}e_{1,0})(x,y)\\ &  \quad +x^{2}(\varphi _{1}\varphi _{2}K_{m,n}e_{0,0})(x,y), \end{align*}
</div>
<div class="displaymath" id="a0000000047">
  \begin{align*}  (\varphi _{1}\varphi _{2}K_{m,n}\varphi _{y}^{2})(x,y) &  =(\varphi _{1}\varphi _{2}K_{m,n}e_{0,2})(x,y)-2y(\varphi _{1}\varphi _{2}K_{m,n}e_{0,1})(x,y)\\ &  \quad +y^{2}(\varphi _{1}\varphi _{2}K_{m,n}e_{0,0})(x,y). \end{align*}
</div>
<p> Next, applying Lemma 3.2. one arrives to <a href="#3.4" class="eqref">3.13</a> and <a href="#3.5" class="eqref">3.14</a>. Suppose now that \(a_{1},b_{1},a_{2},b_{2}\in \mathbb {R}\) such that \(a_{1}{\lt}b_{1}, \  a_{2}{\lt}b_{2}\) and let \(f:[a_{1},b_{1}]\times [a_{2},b_{2}]\to \mathbb {R}\) be a bounded function. </p>
<p>The function \(\omega _{f}:[0,+\infty [\times [0,+\infty [\to \mathbb {R}\) defined by </p>
<div class="displaymath" id="3.6">
  \begin{align} \label{3.6} \omega _{f}(\delta _{1},\delta _{2}) &  =\sup \left\{ |f(x^{\prime },y^{\prime })-f(x^{\prime \prime },y^{\prime \prime })|\; :(x^{\prime },y^{\prime }),(x^{\prime \prime },y^{\prime \prime })\in \right.\\ &  \  \  \  \left.[a_{1},b_{1}]\times [a_{2},b_{2}], \  |x^{\prime }-x^{\prime \prime }|\leq \delta _{1},\  |y^{\prime }-y^{\prime \prime }|\leq \delta _{2}\right\} \nonumber \end{align}
</div>
<p> is called modulus of continuity of the bivariate function \(f\). </p>
<p>Its properties are similar with the properties of the modulus of continuity for univariate functions [3], [7]. </p>
<p>It is known from [7], [24] the following analogous of Shisha-Mond Theorem for the bivariate case: </p>
<p><div class="theorem_thmwrapper " id="a0000000048">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(L:C([a_{1},b_{1}]\times [a_{2},b_{2}])\to B([a_{1},b_{1}]\times [a_{2},b_{2}])\) be a linear positive operator. For any \(f\in C([a_{1},b_{1}]\times [a_{2},b_{2}])\), any \((x,y)\in [a_{1},b_{1}]\times [a_{2},b_{2}]\) and \(\delta _{1}{\gt}0, \  \delta _{2}{\gt}0\) the following inequality </p>
<div class="displaymath" id="3.7">
  \begin{align} \label{3.7} &  |(Lf)(x,y)-f(x,y)|\leq |f(x,y)||(Le_{00})(x,y)-1|+\\ \nonumber &  +\left((Le_{00})(x,y)\! +\! \delta _{1}^{-1}\! \sqrt{(Le_{00})(x,y)(L\varphi _{x}^{2})(x,y)}\! +\! \delta _{2}^{-1}\! \sqrt{(Le_{00})(x,y)(L\varphi _{y}^{2})(x,y)}\right.\\ \nonumber &  \left.+\delta _{1}^{-1}\delta _{2}^{-1}\sqrt{(Le_{00})(x,y)(L\varphi _{x}^{2})(x,y)(L\varphi _{y}^{2})(x,y)}\right)\; \omega _{f}(\delta _{1},\delta _{2})\nonumber \end{align}
</div>
<p> holds. </p>

  </div>
</div> Suppose that the analytic functions \(\varphi _{1},\varphi _{2}:\mathbb {R}\to ]0,+\infty [\) satisfy the conditions </p>
<div class="equation" id="3.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.8} \lim \limits _{y\to \infty }\tfrac {\varphi _{k}^{(1)}(y)}{\varphi _{k}(y)}=\lim \limits _{y\to \infty }\tfrac {\varphi _{k}^{(2)}(y)}{\varphi _{k}(y)}=1 \end{equation}
  </div>
  <span class="equation_label">3.16</span>
</p>
</div>
<p> and taking relation <a href="#3.8" class="eqref">3.16</a> into account, it follows </p>
<div class="displaymath" id="a0000000049">
  \[ \lim \limits _{n\to \infty }\left(\tfrac {\varphi _{k}^{(2)}(nx)}{\varphi _{k}(nx)}-2\tfrac {\varphi _{k}^{(1)}(nx)}{\varphi _{k}(nx)}+1\right)=0, \]
</div>
<p> for \(k\in \{ 1,2\} \). </p>
<p>Then we suppose that, there exists \(0{\lt}\gamma _{k}\leq 1\), so that </p>
<div class="equation" id="3.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.9} \lim \limits _{n\to \infty }n^{\gamma _{k}}\left(\tfrac {\varphi _{k}^{(2)}(nx)}{\varphi _{k}(nx)}-2\tfrac {\varphi _{k}^{(1)}(nx)}{\varphi _{k}(nx)}+1\right)=\beta _{2,k}, \end{equation}
  </div>
  <span class="equation_label">3.17</span>
</p>
</div>
<p> for any \(x\in [0,+\infty [\) and where \(\beta _{2,k}\) are functions, \(\beta _{2,k}:[0,+\infty [\to \mathbb R\), with \(k\in \{ 1,2\} \). </p>
<p><div class="theorem_thmwrapper " id="a0000000050">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any function \(f\in C_{2,2}([0,+\infty [\times [0,+\infty [)\), any bivariate compact interval \([a_{1},b_{1}]\times [a_{2},b_{2}]\subset [0,+\infty [\times [0,+\infty [\), any \((x,y)\in [a_{1},b_{1}]\times [a_{2},b_{2}]\) and \(\delta _{1}{\gt}0\), \(\delta _{2}{\gt}0\), the operators <a href="#3.3" class="eqref">3.12</a> satisfy the following inequality </p>
<div class="displaymath" id="3.10">
  \begin{align} \label{3.10} |(\varphi _{1}\varphi _{2}K_{m,n}f)(x,y)-f(x,y)|\leq K\omega _{f}\left(\tfrac {b_{1}-a_{1}}{\sqrt{m^{\gamma _1}}},\tfrac {b_{2}-a_{2}}{\sqrt{n^{\gamma _2}}}\right), \end{align}
</div>
<p> where </p>
<div class="equation" id="3.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.11} m_{2,k}([a_{k},b_{k}]):=\sup \limits _{x\in [a_{k},b_{k}]}|\beta _{2,k}(x)|, \end{equation}
  </div>
  <span class="equation_label">3.19</span>
</p>
</div>
<p> for \(k\in \{ 1,2\} \) and </p>
<div class="displaymath" id="3.12">
  \begin{align} \label{3.12} K= &  \left(1+\tfrac {1}{b_{1}-a_{1}}\sqrt{\left(m_{2,1}([a_{1},b_{1}])+1\right)b_{1}^{2}+2b_{1}+\tfrac {1}{3}}\right)\\ \nonumber &  \cdot \left(1+\tfrac {1}{b_{2}-a_{2}}\sqrt{\left(m_{2,2}([a_{2},b_{2}])+1\right)b_{2}^{2}+2b_{2}+\tfrac {1}{3}}\right)\nonumber . \end{align}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000051">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>In the Theorem 3.4. one replaces \(L\) by \(\varphi _{1}\varphi _{2}K_{m,n}\) and next, one takes the proof of Theorem 2.8. into account. It is known from [7], [24] the following analogous of Bohman-Korovkin Theorem for the bivariate case: </p>
<p><div class="theorem_thmwrapper " id="a0000000052">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \((L_{m,n})_{m,n\in \mathbb N}\) be a sequence of linear positive operators, \(L_{m,n}:C([a_{1},b_{1}[\times [a_{2},b_{2}[)\to C([a_{1},b_{1}[\times [a_{2},b_{2}[)\) and let \(e_{ij}(x,y)=x^iy^j,\) \(i,j\in \mathbb N_{0}\), \(i+j\leq 2\) be the test functions. </p>
<p>Suppose that the following relations </p>
<ul class="itemize">
  <li><p>\((L_{m,n}e_{00})(x,y)=1,\) </p>
</li>
  <li><p>\((L_{m,n}e_{10})(x,y)=x+u_{m,n}(x,y),\) </p>
</li>
  <li><p>\((L_{m,n}e_{01})(x,y)=y+v_{m,n}(x,y),\) </p>
</li>
  <li><p>\((L_{m,n}(e_{20}+e_{02}))(x,y)=x^2+y^2+w_{m,n}(x,y)\) </p>
</li>
</ul>
<p> hold, for any \((x,y)\in [a_{1},b_{1}]\times [a_{2},b_{2}]\). </p>
<p>If the sequence \((u_{m,n})\), \((v_{m,n})\) and \((w_{m,n})\), \(m,n\in \mathbb N\) converge to zero uniformly on \([a_{1},b_{1}]\times [a_{2},b_{2}]\), then \(L_{m,n}f\) converges to \(f\) uniformly on \([a_{1},b_{1}]\times [a_{2},b_{2}]\), for any \(f\in C([a_{1},b_{1}]\times [a_{2},b_{2}])\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000053">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose that the analytic functions \(\varphi _{1},\varphi _{2}:\mathbb {R}\to ]0,+\infty [\) satisfy the conditions <a href="#3.8" class="eqref">3.16</a>. </p>
<p>Then, for every function \(f\in C_{2,2}([0,+\infty [\times [0,+\infty [)\) it holds </p>
<div class="displaymath" id="a0000000054">
  \[ \lim \limits _{m,n\to \infty }(\varphi _{1}\varphi _{2}K_{m,n}f)(x,y)=f(x,y) \]
</div>
<p> uniformly on any bivariate compact interval \([a_{1},\! b_{1}]\times [a_{2},b_{2}]\! \subset [0,\! +\infty [\times [0,\! +\infty [.\) </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000055">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Applying Lemma 3.2. and the hypothesis <a href="#3.8" class="eqref">3.16</a>, it follows </p>
<div class="displaymath" id="a0000000056">
  \[ \lim \limits _{m,n\to \infty }(\varphi _{1}\varphi _{2}K_{m,n}e_{10})(x,y)=x, \]
</div>
<div class="displaymath" id="a0000000057">
  \[ \lim \limits _{m,n\to \infty }(\varphi _{1}\varphi _{2}K_{m,n}e_{01})(x,y)=y \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000058">
  \[ \lim \limits _{m,n\to \infty }\big(\varphi _{1}\varphi _{2}K_{m,n}(e_{20}+e_{02})\big)(x,y)=x^2+y^2 \]
</div>
<p> uniformly on any bivariate compact interval \([a_{1},b_{1}]\times [a_{2},b_{2}]\! \subset [0,\! +\infty [\times [0,\! +\infty [\). </p>
<p>Next, using Theorem 3.6. one arrives to the desired result. </p>
<h1 id="a0000000059">4 The GBS \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operator</h1>
<p>In this section we shall construct the \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operator associated to a B-continuous function. </p>
<p>A function \(f:I\times J\to \mathbb R\) is called B-continuous function in \((x_{0},y_{0})\in I\times J\), if </p>
<div class="displaymath" id="a0000000060">
  \[ \lim \limits _{(x,y)\to (x_{0},y_{0})}\Delta f[(x,y),(x_{0},y_{0})]=0, \]
</div>
<p> where </p>
<div class="equation" id="4.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.1} \Delta f[(x,y),(x_{0},y_{0})]=f(x,y)-f(x_{0},y)-f(x,y_{0})+f(x_{0},y_{0}) \end{equation}
  </div>
  <span class="equation_label">4.20</span>
</p>
</div>
<p> denotes a so-called mixed difference of function \(f\). A function \(f:I\times J\to \mathbb R\) is called B-continuous function on \(I\times J\), if it is B-continuous at any point of \(I\times J\). The definition of B-continuity was introduced by K. Bögel in [10]. The function \(f:I\times J\to \mathbb R\) is B-bounded on \(I\times J\), if there exists \(k{\gt}0\) so that </p>
<div class="displaymath" id="a0000000061">
  \[  |\Delta f[(x,y),(s,t)]|\leq k, \  \mbox{for any } ((x,y),(s,t))\in I\times J.  \]
</div>
<p>We shall use the function sets:<br />\(B(I\times J)=\left\{ f|\; f:I\times J\to \mathbb R,\; f\right.\) is bounded on \(\left.I\times J\right\} \),<br />with the usual sup-norm \(\| \cdot \| _{\infty }\), <br />\(B_{b}(I\times J)=\left\{ f|\; f:I\times J\to \mathbb R,\;  f\right.\) is B-bounded on \(\left.I\times J\right\} ,\)<br />\(C_{b}(I\times J)=\left\{ f|\; f:I\times J\to \mathbb R,\;  f\right.\) is B-continuous on \(\left.I\times J\right\} .\)<br />Let \(f\in B_{b}(I\times J)\). The function \(\omega _{mixed}(f;\ast ,\ast ):[0,+\infty [\times [0,+\infty [\to \mathbb R\) defined by </p>
<div class="equation" id="4.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.2} \omega _{mixed}(f;\delta _{1},\delta _{2}):=\sup \{ |\Delta f[(x,y),(s,t)]|:|x-s|\leq \delta _{1},\; |y-t|\leq \delta _{2}\} , \end{equation}
  </div>
  <span class="equation_label">4.21</span>
</p>
</div>
<p> for any \((\delta _{1},\delta _{2})\in [0,+\infty [\times [0,+\infty [\) is called the mixed modulus of smoothness. The notion of mixed modulus of smoothness was introduced by I. Badea in [3] and then studied by I. Badea, C. Badea, C. Cottin and H. H. Gonska [4], [6]. </p>
<p><div class="definition_thmwrapper " id="a0000000062">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">4.1</span>
  </div>
  <div class="definition_thmcontent">
  <p><em>[4]</em> Let \(I,J\subset \mathbb {R}\) be compact intervals and let \(L:C_{b}(I\times J)\to B(I\times J)\) be a linear positive operator. Suppose that \(f(\cdot ,\ast )\in C_{b}(I\times J)\). The operator \(U:C_{b}(I\times J)\to B(I\times J)\) defined for any \(f\in C_{b}(I\times J)\) and \((x,y)\in I\times J\) by </p>
<div class="equation" id="4.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.3} (Uf)(x,y)=L\left(f(\cdot ,y)+f(x,\ast )-f(\cdot ,\ast )\right)(x,y) \end{equation}
  </div>
  <span class="equation_label">4.22</span>
</p>
</div>
<p> is called the GBS (Generalized Boolean Sum) operator associated to \(L\). </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000063">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>The notion of GBS operator was introduced by C. Badea and C. Cottin [4].<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000064">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>The most natural way to construct the GBS operator \(U\) is the following: </p>
<ul class="itemize">
  <li><p>one considers the univariate operators </p>
</li>
</ul>
<div class="displaymath" id="a0000000065">
  \[ L_{1}^{x}:C_{b}(I)\to B(I), \  \  L_{2}^{y}:C_{b}(J)\to B(J); \]
</div>
<ul class="itemize">
  <li><p>if </p>
</li>
</ul>
<p> \(L_{1}^{x},L_{2}^{y}:C_{b}(I\times J)\to B(I\times J)\) are their parametric extensions [7], [12], then </p>
<div class="equation" id="4.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.4} U=L_{1}^{x}+L_{2}^{y}-L_{1}^{x}L_{2}^{y}. \end{equation}
  </div>
  <span class="equation_label">4.23</span>
</p>
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000066">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4.4</span>
  </div>
  <div class="lemma_thmcontent">
  <p>The GBS operator associated to the bivariate \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operators is defined for any \(f:[0,+\infty [\times [0,+\infty [\to \mathbb {R}\) and \((x,y)\in [0,+\infty [\times [0,+\infty [\) by </p>
<div class="displaymath" id="4.5">
  \begin{align} \label{4.5} &  (\varphi _{1}\varphi _{2}U_{m,n}f)(x,y)=\tfrac {m}{\varphi _{1}(mx)}\tfrac {n}{\varphi _{2}(ny)}\sum \limits _{k=0}^{\infty }\sum \limits _{j=0}^{\infty }\tfrac {\varphi _{1}^{(k)}(0)}{k!}\tfrac {\varphi _{2}^{(j)}(0)}{j!}\\ &  \  \  \  \cdot (mx)^{k}(ny)^{j}\int _{\tfrac {k}{m}}^{\tfrac {k+1}{m}}\int _{\tfrac {j}{n}}^{\tfrac {j+1}{n}}\left(f\left(\tfrac {k}{m},t\right)+f\left(s,\tfrac {j}{n}\right)-f\left(\tfrac {k}{m},\tfrac {j}{n}\right)\right){\rm d}s{\rm d}t.\nonumber \end{align}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000067">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>One applies relation <a href="#4.4" class="eqref">4.23</a> with \(L_{1}^{x}:=\varphi _{1}S_{m}^{x}\), \(L_{2}^{y}:=\varphi _{2}S_{n}^{y}\) and one takes Lemma 3.1. into account. The analogous of Shisha-Mond Theorem in terms of mixed modulus of smoothness is the following: </p>
<p><div class="theorem_thmwrapper " id="a0000000068">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p><em>[4]</em> Let \(L:C_{b}(I\times J)\to B(I\times J)\) be an linear and positive operator reproducing constants and let \(U:C_{b}(I\times J)\to B(I\times J)\) be the GBS associated operator. </p>
<p>For any \((x,y)\in I\times J\) and \((\delta _{1},\delta _{2})\in ]0,+\infty [\times ]0,+\infty [\) the following inequality: </p>
<div class="displaymath" id="4.6">
  \begin{align} \label{4.6} |f(x,y)-(Uf)(x,y)| &  \leq \left(1+\delta _{1}^{-1}\sqrt{(L\varphi _{x}^{2})(x,y)}+\delta _{2}^{-1}\sqrt{(L\varphi _{y}^{2})(x,y)}\right.\\ &  \quad \left.+\delta _{1}^{-1}\delta _{2}^{-1}\sqrt{(L\varphi _{x}^{2})(x,y)(L\varphi _{y}^{2})(x,y)}\right)\omega _{mixed}(f;\delta _{1},\delta _{2})\nonumber \end{align}
</div>
<p> holds. </p>

  </div>
</div> For the GBS operator associated to the \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operator we have the following: </p>
<p><div class="theorem_thmwrapper " id="a0000000069">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any function \(f\in C_{b}([0,+\infty [\times [0,+\infty [)\), any bivariate interval \([a_{1},b_{1}]\times [a_{2},b_{2}]\subset [0,+\infty [\times [0,+\infty [\) and \((x,y)\in [a_{1},b_{1}]\times [a_{2},b_{2}]\), the GBS operator <a href="#4.5" class="eqref">4.24</a> satisfies the following inequality </p>
<div class="equation" id="4.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.7} |f(x,y)-(\varphi _{1}\varphi _{2}U_{m,n}f)(x,y)|\leq K\omega _{mixed}\left(f;\tfrac {b_{1}-a_{1}}{\sqrt{m^{\gamma _1}}},\tfrac {b_{2}-a_{2}}{\sqrt{n^{\gamma _2}}}\right), \end{equation}
  </div>
  <span class="equation_label">4.26</span>
</p>
</div>
<p> where K is defined at <a href="#3.11" class="eqref">3.19</a> and <a href="#3.12" class="eqref">3.20</a>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000070">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>One applies the Theorem 4.5. with \(L:=\varphi _{1}\varphi _{2}K_{m,n},\) respectively with \(U:=\varphi _{1}\varphi _{2}U_{m,n}\). Next we recall the Korovkin type theorem for B-continuous functions due to C. Badea, I. Badea and H. H. Gonska in [5]. </p>
<p><div class="theorem_thmwrapper " id="a0000000071">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p><em>[5]</em> Let \((L_{m,n})_{m,n\in \mathbb N}\) be a sequence of linear positive operators, \(L_{m,n}:C_{b}(I\times J)\to B(I\times J)\). If </p>
<ul class="itemize">
  <li><p>\((L_{m,n}e_{00})(x,y)=1,\) </p>
</li>
  <li><p>\((L_{m,n}e_{10})(x,y)=x+u_{m,n}(x,y),\) </p>
</li>
  <li><p>\((L_{m,n}e_{01})(x,y)=y+v_{m,n}(x,y),\) </p>
</li>
  <li><p>\((L_{m,n}(e_{20}+e_{02}))(x,y)=x^2+y^2+w_{m,n}(x,y),\) </p>
</li>
  <li><p>\(\lim \limits _{m,n\to \infty }u_{m,n}(x,y)=\lim \limits _{m,n\to \infty }v_{m,n}(x,y)=\lim \limits _{m,n\to \infty }w_{m,n}(x,y)=0\)<br />uniformly on \(I\times J\), </p>
</li>
</ul>
<p> then for any \(f\in C_{b}(I\times J)\), any \((x,y)\in I\times J\), the sequence \((U_{m,n})_{m,n\in \mathbb N}\) converges to \(f\) uniformly on \(I\times J\), where the operator \(U_{m,n}\), \(m,n\in \mathbb N\) is defined by </p>
<div class="equation" id="4.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.8} (U_{m,n}f)(x,y)=L_{m,n}(f(\cdot ,y)+f(x,\ast )-f(\cdot ,\ast ))(x,y). \end{equation}
  </div>
  <span class="equation_label">4.27</span>
</p>
</div>

  </div>
</div> For the GBS operator associated to the \(\varphi _{1}\varphi _{2}\)-Szász-Mirakjan-Kantorovich operator we have the following: </p>
<p><div class="theorem_thmwrapper " id="a0000000072">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose that the analytic functions \(\varphi _{1}, \varphi _{2}:\mathbb R\to ]0,+\infty [\) satisfy the conditions <a href="#3.8" class="eqref">3.16</a>. Then, for any function \(f\in C_{b}([0,+\infty [\times [0,+\infty [)\) and any \((x,y)\in [a_{1},b_{1}]\times [a_{2},b_{2}]\), it holds </p>
<div class="displaymath" id="a0000000073">
  \[ \lim \limits _{m,n\to \infty }(\varphi _{1}\varphi _{2}U_{m,n}f)(x,y)=f(x,y) \]
</div>
<p> uniformly on any bivariate compact interval \([a_{1},b_{1}]\times [a_{2},\! b_{2}]\subset [0,\! +\infty [\times [0,\! +\infty [\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000074">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> One applies Theorem 4.7. with \(L_{m,n}:=\varphi _{1}\varphi _{2}K_{m,n}\), respectively with \(U_{m,n}:=\varphi _{1}\varphi _{2}U_{m,n}\). <div class="proof_wrapper" id="a0000000075">
  <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>
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</dd>
  <dt><a name="222">2</a></dt>
  <dd><p><i class="sc">Altomare, F.</i> and <i class="sc">Campiti, M.</i>, <i class="it">Korovkin-type Approximation Theory and Its Applications</i>, de Gruyter Series in Mathematics, <b class="bf">17</b>, Walter de Gruyter &amp; Co., Berlin, New York, 1994. </p>
</dd>
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</dd>
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</dd>
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</dd>
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  <dd><p><i class="sc">Ciupa, A.</i> and <i class="sc">Gavrea, I.</i>, <em>On a Favard-Szász type operator</em>, Studia Univ. “Babeş-Bolyai", Mathematica, <b class="bfseries">34</b>, pp.&#160;39–46, 1994. </p>
</dd>
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  <dd><p><i class="sc">Delvos, F. J.</i> and <i class="sc">Schempp, W.</i>, <em>Boolean Methods in Interpolation and Approximation</em>, Longman Scientific and Technical, 1989. </p>
</dd>
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</dd>
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  <dd><p><i class="sc">Favard, J.</i>, <i class="it">Sur les multiplicateurs d’interpolation</i>, Journal Pures Appl., <b class="bf">23</b> (9), pp. 219–247, 1944. </p>
</dd>
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  <dd><p><i class="sc">Jakimovski, A.</i> and <i class="sc">Leviatan, D.</i>, <i class="it">Generalized Szász operators for the approximation in the infinite interval</i>, Mathematica (Cluj), <b class="bf">34</b>, pp. 97–103, 1969. </p>
</dd>
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  <dd><p><i class="sc">Kantorovich, L. V.</i>, <em>Sur certain développements suivant les polynômes de la forme de S. Bernstein</em>, <b class="bf">I</b>, <b class="bf">II</b>, C. R. Acad. URSS, pp.&#160;563–568, 595–600, 1930. </p>
</dd>
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  <dd><p><i class="sc">Miclăuş, D.</i>, <i class="it">The Voronovskaja type theorem for the Szász-Mirakjan-Kantorovich operators</i>, Journal of Science and Arts, <b class="bf">2</b> (13), pp. 257–260, 2010. </p>
</dd>
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  <dd><p><i class="sc">Mirakjan, G. M.</i>, <em>Approximation of continuous functions with the aid of polynomials</em>, Dokl. Acad. Nauk SSSR, <b class="bf">31</b>, pp.&#160;201–205, 1941. </p>
</dd>
  <dt><a name="16">18</a></dt>
  <dd><p><i class="sc">Mortici, C.</i>, <em>An Extension of the Szász-Mirakjan Operators</em>, An. Şt. Univ. Ovidius Constanţa, <b class="bfseries">17</b> (1), pp.&#160;137–144, 2009. </p>
</dd>
  <dt><a name="17">19</a></dt>
  <dd><p><i class="sc">Pop, O. T., Bărbosu, D.</i> and <i class="sc">Miclăuş, D.</i>, <em>The Voronovskaja type theorem for an extension of Szász-Mirakjan operators</em>, Demonstratio Mathematica, to appear. </p>
</dd>
  <dt><a name="18">20</a></dt>
  <dd><p><i class="sc">Shisha, O.</i> and <i class="sc">Mond, B.</i>, <em>The degree of convergence of linear positive operators</em>, Proc. Nat. Acad. Sci. U.S.A., <b class="bfseries">60</b>, pp.&#160;1196–1200, 1968. </p>
</dd>
  <dt><a name="19">21</a></dt>
  <dd><p><i class="sc">Stancu, D. D.</i>, <em>Approximation of function by a new class of linear polynomial operators</em>, Rev. Roum. Math. Pures et Appl., <b class="bfseries">13</b>, pp.&#160;1173–1194, 1968. </p>
</dd>
  <dt><a name="20">22</a></dt>
  <dd><p><i class="sc">Stancu, D. D., Coman, Gh., Agratini, O.</i> and <i class="sc">Trâmbiţaş, R.</i>, <em>Numerical Analysis and Approximation Theory</em>, <b class="bf">I</b>, Presa Universitară Clujeană, Cluj-Napoca, 2001 (in Romanian). </p>
</dd>
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  <dd><p><i class="sc">Stancu, F.</i>, <em>On the remainder term in approximation formula by univariate and bivariate Mirakjan operators</em>, An. Şt. Univ. “Al. I. Cuza" Iaşi, <b class="bfseries">XIV</b>, pp.&#160;415–422, 1968 (in Romanian). </p>
</dd>
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</dd>
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  <dd><p><i class="sc">Totik, V.</i>, <em>Uniform approximation by positive operators on infinite intervals</em>, Analysis Mathematica, <b class="bfseries">10</b>, pp.&#160;163–182, 1984. </p>
</dd>
</dl>


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