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<title>On the rate of convergence <br />of modified \(\alpha \)-Bernstein operators based on \(q\)-integers : On the rate of convergence <br />of modified \(\alpha \)-Bernstein operators based on \(q\)-integers </title>
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<h1>On the rate of convergence <br />of modified \(\alpha \)-Bernstein operators based on \(q\)-integers </h1>
<p class="authors">
<span class="author">Purshottam N. Agrawal,\(^\ast \) Dharmendra Kumar\(^\ast \) Behar Baxhaku\(^{\ast \ast }\)</span>
</p>
<p class="date">October 8, 2021; accepted: December 29, 2021; published online: August 25, 2022.</p>
</div>
<div class="abstract"><p> In the present paper we define a q-analogue of the modified \(\alpha \)-Bernstein operators introduced by Kajla and Acar (<a href ="https://doi.org/10.1215/20088752-2019-0015"><i class="it">Ann. Funct. Anal.</i>, 10 (2019), 570–582</a>). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre’s \(K\)-functional. Next, we investigate the Grüss-Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre’s \(K\)-functional, the Voronovskaja and Grüss-Voronovskaja type theorems are also investigated. Finally, we construct the associated <i class="it">GBS</i> (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness. </p>
<p><b class="bf">MSC.</b> 41A25, 41A36, 41A63, 41A10. </p>
<p><b class="bf">Keywords.</b> Steklov mean, Peetre’s K-functional, modulus of continuity, Lipschitz class, Voronovskaja type theorem, Grüss-Voronovskaja type theorem, mixed modulus of smoothness. </p>
</div>
<p>\(^\ast \)Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India, e-mails: <span class="tt">pnappfma@gmail.com</span>, <span class="tt">dharmendrak.dav@gmail.com</span>. </p>
<p>\(^{\ast \ast }\)Department of Mathematics, University of Prishtina, Prishtina, Kosovo, e-mail: <span class="tt">behar.baxhaku@uni-pr.edu</span>, corresponding author. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>In 1912, Bernstein <span class="cite">
	[
	<a href="#BEB" >9</a>
	]
</span> gave a simple and elegant proof of the Weierstrass approximation theorem by defining a sequence of positive linear operators as follows. </p>
<p>For \(\zeta \in \mathcal{C}(\mathcal{I})=\{ f:\mathcal{I}\rightarrow \mathbb {R}|\;  f\;  is\;  continuous\} \), \(\mathcal{I}=[0,1]\), the \(n\)-th Bernstein polynomial \(\mathcal{B}_n:\mathcal{C}(\mathcal{I})\rightarrow \mathcal{C}(\mathcal{I})\) is defined by </p>
<div class="equation" id="F1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{F1.1} \mathcal{B}_n(\zeta ;x)=\sum ^n_{k=0}\textstyle {n\choose k}x^{k}(1-x)^{n-k}\zeta \big(\tfrac {k}{n}\big),\qquad \forall x\in \mathcal{I}~  and~  n\in \mathbb {N}. \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p>Later several authors introduced various generalizations of these polynomials and studied their approximation properties (see <span class="cite">
	[
	<a href="#GA" >22</a>
	]
</span>). </p>
<p>In the past two decades, the development of \(q\)-calculus has been an active area of research. In \(1987\), Lupas <span class="cite">
	[
	<a href="#AL" >29</a>
	]
</span> was the first person who introduced a \(q\)-analogue of the Bernstein polynomials and established some approximation results. A decade later, Phillips <span class="cite">
	[
	<a href="#GM" >35</a>
	]
</span> gave another \(q\)-analogue, \(\mathcal{L}_{n,q}:\mathcal{C}(\mathcal{I})\rightarrow \mathcal{C}(\mathcal{I})\) of the Bernstein polynomials which became very popular. He defined it as </p>
<div class="equation" id="eq1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq1.1} \mathcal{L}_{n,q}(\zeta ;x)=\sum ^n_{k=0}\zeta \left(\tfrac {[k]_q}{[n]_q}\right)\mathfrak {p}_{n,k}(q;x), \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p> where \(0{\lt}q{\lt}1\) and \(\mathfrak {p}_{n,k}(q;x)={n\choose k}_{q}x^{k}(1-x)^{n-k}_{q},\; x\in \mathcal{I}.\) Several generalizations have been studied for the definition of q-Bernstein polynomials given by (<a href="#eq1.1">1.2</a>), for instance, Muraru <span class="cite">
	[
	<a href="#Muraru" >31</a>
	]
</span> proposed a \(q\)-analogue of the Bernstein-Schurer operators and examined convergence behaviour by virtue of the modulus of continuity. Agrawal <i class="it">et al.</i> <span class="cite">
	[
	<a href="#AGK" >2</a>
	]
</span> defined Bernstein-Schurer-Stancu operators based on \(q\)-integers and discussed the local and global results. Dalmanog̎lu <span class="cite">
	[
	<a href="#OD" >33</a>
	]
</span> gave a \(q\)-analogue of the Bernstein-Kantorovich polynomials. Gupta <span class="cite">
	[
	<a href="#GA" >22</a>
	]
</span> introduced a sequence of Bernstein-Durrmeyer operators based on \(q\)-integers and established some approximation theorems. </p>
<p>Gupta and Radu <span class="cite">
	[
	<a href="#AG" >24</a>
	]
</span> discussed statistical approximation properties of another kind of \(q\)-Baskakov-Kantorovich operators. Mahmudov <span class="cite">
	[
	<a href="#NIM" >30</a>
	]
</span> introduced an alternate form of the \(q\)-analogue of Bernstein-Kantorovich operators considered in <span class="cite">
	[
	<a href="#OD" >33</a>
	]
</span>. Aliaga and Baxhaku <span class="cite">
	[
	<a href="#AB" >4</a>
	]
</span> defined a bivariate extension of the \(q\)-Bernstein type operators involving parameter \(\lambda \) and examined their degree of approximation. </p>
<p>Recently, Mursaleen <i class="it">et al.</i> <span class="cite">
	[
	<a href="#MMA" >32</a>
	]
</span> proposed a sequence of generalized Bernstein operators based on \(q\)-integers. Cai <span class="cite">
	[
	<a href="#CX" >12</a>
	]
</span> introduced another generalization of Bernstein operators based on \(q\)-integers and derived some convergence theorems and shape preserving properties. Based on the Phillips \(q\)-Bernstein polynomials <span class="cite">
	[
	<a href="#GM" >35</a>
	]
</span>, generalized Bézier curves and surfaces were introduced in (<span class="cite">
	[
	<a href="#DHA" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#DOH" >16</a>
	]
</span>, <span class="cite">
	[
	<a href="#OHP" >34</a>
	]
</span>). For a detailed account of the work in this direction, we refer the reader to the book <span class="cite">
	[
	<a href="#AGA" >5</a>
	]
</span>. Before proceeding further, let us mention some important basic definitions and notations of \(q\)-calculus. For \(q{\gt}0\), and each nonnegative integer \(l\), the q-integer \([l]_q\) and the q-factorial \([l]_q!\) are, respectively, given by </p>
<div class="displaymath" id="a0000000003">
  \[  [l]_{q}=\left\{ \begin{array}{ll} \frac{(1-q^{l})}{(1-q)},&  q\neq 1,\\[2mm] l,&  q=1. \end{array}\right.  \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000004">
  \[  [l]_{q}!=\left\{ \begin{array}{ll} [l]_{q}[l-1]_q \ldots [1]_q,&  l\geq 1,\\[2mm] 1,&  l=0. \end{array}\right.  \]
</div>
<p> For the non-negative integers \(n,l\) satisfying \(l\leq n\), the q-binomial coefficients are defined by </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  \textstyle {n\choose l}_{q} :=\tfrac {[n]_q!}{[l]_q![n-l]_q!}. \end{equation*}
</div>
<p>In 2017, Chen <i class="it">et al.</i> <span class="cite">
	[
	<a href="#Chen" >13</a>
	]
</span> introduced generalized Bernstein operators (<a href="#F1.1">1.1</a>) involving a real parameter ‘\(\alpha \)’ satisfying \(0\leq \alpha \leq 1\), as </p>
<div class="equation" id="F1.04">
<p>
  <div class="equation_content">
    \begin{equation} \label{F1.04} \mathcal{L}_{n,\alpha }(\zeta ;x)=\sum _{k=0}^n P_{n,k,\alpha }(x)\zeta \big(\tfrac {k}{n}\big), \;  x\in \mathcal{I}, \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> where the \(\alpha \)-Bernstein basis function \(P_{n,k,\alpha }(x),\) for \(n\geq 2,\) is given by </p>
<div class="displaymath" id="F1.3">
  \begin{align} \label{F1.3} P_{n,k,\alpha }(x)=& \textstyle {n-2\choose k} (1-\alpha ){x}^k(1-x)^{n-k-1}\\ & +\textstyle {n-2\choose k-2} (1-\alpha ){x}^{k-1}(1-x)^{n-k}\nonumber \\ & +\textstyle {n\choose k}\alpha {x}^k(1-x)^{n-k}\nonumber \end{align}
</div>
<p> with \({n-2\choose -2}={n-2\choose -1}=0\). Clearly \(P_{n,k,\alpha }(x),\) verifies the following recurrence relation </p>
<div class="displaymath" id="a0000000006">
  \begin{eqnarray*}  P_{n,k,\alpha }(x)=(1-x)P_{n-1,k,\alpha }(x)+x P_{n-1,k-1,\alpha }(x),\; \forall \; \; 0{\lt}k{\lt}n, \;  and\; \;  n\geq 3. \end{eqnarray*}
</div>
<p>The authors <span class="cite">
	[
	<a href="#Chen" >13</a>
	]
</span> established the uniform convergence theorem and the Voronovskaja-type asymptotic formula etc. For the special case \(\alpha =1\), the operators (<a href="#F1.04">1.3</a>) reduce to (<a href="#F1.1">1.1</a>). </p>
<p>For any \(\zeta \in \mathcal{C}(\mathcal{I}),\) Khosravian-Arab <i class="it">et al.</i> <span class="cite">
	[
	<a href="#Khosravian" >27</a>
	]
</span> presented a new family of Bernstein operators as follows: </p>
<div class="displaymath" id="F1.07">
  \begin{eqnarray} \label{F1.07} B_n^{M,1}(\zeta ;x)& =& \sum _{m=0}^n P_{n,m}^{M,1}(x)\zeta \big(\tfrac {m}{n}\big),\; \;  x\in \mathcal{I}, \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="a0000000007">
  \begin{align*}  P_{n,m}^{M,1}(x)& =a(x,n)P_{n-1,m}^{M,1}(x)+a(1-x,n)P_{n-1,m-1}^{M,1}(x),\; \;  1\leq m\leq n-1,\nonumber \\ P_{n,0}^{M,1}(x)& =a(x,n)(1-x)^{n-1},\nonumber \\ P_{n,n}^{M,1}(x)& =a(1-x,n){x}^{n-1},\nonumber \end{align*}
</div>
<p> and </p>
<div class="equation" id="F1.03">
<p>
  <div class="equation_content">
    \begin{equation} \label{F1.03} a(x,n)=a_0(n)+x\; a_1(n),\; \; n=0,1,2,3\ldots \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> \(a_0(n)\) and \(a_1(n)\) being two unknown sequences, which may be defined in an appropriate manner. If \(a_0(n)=1,\) and \(a_1(n)=-1,\) then (<a href="#F1.07">1.5</a>) reduces to (<a href="#F1.1">1.1</a>). </p>
<p>Recently, Kajla and Acar <span class="cite">
	[
	<a href="#KT" >26</a>
	]
</span> for any \(\zeta \in \mathcal{C}(\mathcal{I}),\) constructed a new family of \(\alpha \)-Bernstein operators defined by </p>
<div class="equation" id="je1">
<p>
  <div class="equation_content">
    \begin{equation} \label{je1} \mathcal{G}^{M,1}_{n,\alpha }(\zeta ;x)=\sum ^n_{k=0}P^{M,1}_{n,k,\alpha }(x)\zeta \left(\tfrac {k}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p> where \(P^{M,1}_{n,k,\alpha }(x)=a(x,n)P_{n-1,k,\alpha }(x)+a(1-x,n)P_{n-1,k-1,\alpha }(x)\) and \(a(x,n)=a_1(n)x+a_0(n)\), and investigated some approximation properties such as Korovkin type theorem and a Voronovskaja type asymptotic formula. Clearly, for \(a_0(n)=1,\) and \(a_1(n)=-1,\) (<a href="#je1">1.4</a>) includes (<a href="#F1.04">1.3</a>). </p>
<p>In 2019, Gupta <i class="it">et al.</i> <span class="cite">
	[
	<a href="#V Gupta" >23</a>
	]
</span> presented a Kantorovich variant of the operators (<a href="#F1.07">1.5</a>) and determined various approximation results. </p>
<p>Motivated by the above research, for \(\zeta \in \mathcal{C}(\mathcal{I})\) endowed with the uniform norm \(\| .\| \), we define the \(q\)-analogue of the operators \(\eqref{je1}\) as follows: </p>
<div class="equation" id="je2">
<p>
  <div class="equation_content">
    \begin{equation} \label{je2} \mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q;x)=\sum ^n_{k=0}P^{M,1}_{n,k,\alpha }(q;x)\zeta \left(\tfrac {[k]_q}{[n]_q}\right), \end{equation}
  </div>
  <span class="equation_label">1.5</span>
</p>
</div>
<p> where \(P^{M,1}_{n,k,\alpha }(q;x)=a(x,n,q)P_{n-1,k,\alpha }(q;x)+a(1-x,n,q)P_{n-1,k-1,\alpha }(q,x)\) with \(P_{n,k,\alpha }(q;x)={n-2\choose k}_{q}(1-\alpha )(qx)^{k}(1-x)_{q}^{n-k-1}+{n-2\choose k-2}_{q}(1-\alpha )q^{k-2}x^{k-1}(1-qx)_{q}^{n-k}+{n\choose k}_{q} \alpha (qx)^{k}(1-qx)_{q}^{n-k}\) and \(a(x,n,q)=a_1(n,q)x+a_0(n,q)\). For \(a_1(n,q)=-1\) and \(a_0(n,q)=1\) and \(\alpha =1\), we get \(q\)-Bernstein polynomials given by (<a href="#eq1.1">1.2</a>). </p>
<p>The purpose of the given article is to examine the convergence behavior of the operators \(\eqref{je2}\) by virtue of the Lipschitz class and the \(K\)-functional. Next, we study a bivariate extension of these operators and determine the convergence estimates by virtue of the moduli of continuity and the \(K\)-functional. Further, we study the Voronovskaja and Grüss Voronovskaja type theorems and establish a quantitative result for functions in the Lipschitz class. Lastly, we extend our study to the corresponding GBS operators. </p>
<h1 id="a0000000008">2 Preliminaries</h1>
<p> Throughout this paper, we assume that \(2a_0(n,q)+a_1(n,q)=1\). <div class="lemma_thmwrapper " id="l1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(e_j(\hslash )=\hslash ^{j}, j=0,1,2,\) the moments of the operators \(\eqref{je2}\), are given by </p>
<ul class="itemize">
  <li><p>\(\mathcal{G}^{M,1}_{n,\alpha }(e_0;q;x)=1,\) </p>
</li>
  <li><p>\(\mathcal{G}^{M,1}_{n,\alpha }(e_1;q;x)=\tfrac {1}{[n]_q}\bigg[a(x,n,q)\{ (1-\alpha )[n-3]_qqx(1-x)+(1-\alpha )x([2]_q+q^3[n-3]_qx)+\alpha [n-1]_q qx\}  +a(1-x,n,q)\{ (1-\alpha )(1-x)(1+q^2[n-3]_qx)+(1-\alpha )x([3]_q+q^4[n-3]_qx)+\alpha (1+q^2[n-1]_qx)\} \bigg],\) </p>
</li>
  <li><p>\(\mathcal{G}^{M,1}_{n,\alpha }(e_2,q;x)=\tfrac {a(x,n,q)}{[n]_q^2}\bigg[(1-\alpha )[n-3]_qqx(1-x)+(1-\alpha )[n-3]_q[n-4]_qq^3x^2(1-x)+(1-\alpha )x[2]^2_q +2[2]_q(1-\alpha )q^3x^2[n-3]_q+(1-\alpha )q^5x^2[n-3]_q+(1-\alpha )q^7x^3[n-3]_q[n-4]_q+\alpha [n-1]_qqx+\alpha [n-1]_q[n-2]_qq^3x\}  +\tfrac {a(1-x,n,q)}{[n]_q^2}\bigg[(1-\alpha )(1-x)+2(1-\alpha )q^2x(1-x)[n-3]_q+(1-\alpha )q^3x(1-x)[n-3]_q+(1-\alpha )q^5x^2(1-x)[n-3]_q[n-4]_q +(1-\alpha )x[3]^2_q+2(1-\alpha )[3]_qq^4x^2[n-3]_q+q^7(1-\alpha )x^2[n-3]_q+q^9(1-\alpha )x^3[n-3]_q[n-4]_q +\alpha +2q^2\alpha x[n-1]_q+\alpha q^3x[n-1]_q+\alpha q^5x^2[n-1]_q[n-2]_q\bigg].\) </p>
</li>
</ul>

  </div>
</div> From the above lemma, one can obtain: <div class="lemma_thmwrapper " id="l2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The central moments of the operators \((\ref{je2})\), for \(\phi ^{j}_{x}(\hslash )=(\hslash -x)^{j}\), where j=1,2 are given by </p>
<ul class="itemize">
  <li><p>\(\mathcal{G}^{M,1}_{n,\alpha }(\phi ^{1}_{x};q;x)=\tfrac {1}{[n]_q}\bigg[a(x,n,q)\{ (1-\alpha )[n-3]_qqx(1-x)+(1-\alpha )x([2]_q+q^3[n-3]_qx)+\alpha [n-1]_q qx\}  +a(1-x,n,q)\{ (1-\alpha )(1-x)(1+q^2[n-3]_qx)+(1-\alpha )x([3]_q+q^4[n-3]_qx)+\alpha (1+q^2[n-1]_qx)\} -[n]_qx\bigg],\) </p>
</li>
  <li><p>\(\mathcal{G}^{M,1}_{n,\alpha }(\phi ^{2}_{x};q;x)=\tfrac {a(x,n,q)}{[n]_q^2}\bigg[(1-\alpha )[n-3]_q2x(1-x)+(1-\alpha )[n-3]_q[n-4]_qq^3x^2(1-x)+(1-\alpha )x[2]_q^2+2[2]_q(1-\alpha )q^3x^2[n-3]_q+(1-\alpha )q^5x^2[n-3]_q+(1-\alpha )q^7x^3[n-3]_q[n-4]_q+\alpha [n-1]_qqx+\alpha [n-1]_q[n-2]_qq^3x^2-2x^2(1-x)q(1-\alpha )[n-3]_q[n]_q-2(1-\alpha )x^2[n]_q([2]_q+q^3[n-3]_qx)-2\alpha [n]_q[n-1]_qqx^2+[n]_q^2x^2\bigg] +\tfrac {a(1-x,n,q)}{[n]_q^2}\bigg[(1-\alpha )(1-x)+2(1-\alpha )q^2x(1-x)[n-3]_q+(1-\alpha )q^3x(1-x)[n-3]_q+(1-\alpha )q^5x^2(1-x)[n-3]_q[n-4]_q+(1-\alpha )x[3]_q^2+2(1-\alpha )[3]_qq^4x^2[n-3]_q+q^7(1-\alpha )x^2[n-3]_q+q^9(1-\alpha ) x^3[n-3]_q[n-4]_q+\alpha +2q^2\alpha x[n-1]_q+\alpha q^3x[n-1]_q+\alpha q^5x^2[n-1]_q[n-2]_q-2x[n]_q(1-\alpha )(1-x)(1+q^2[n-3]_qx)-2x^2(1-\alpha )([3]_q+q^4[n-3]_qx)[n]_q-2x\alpha (1+q^2[n-1]_qx)[n]_q+[n]_q^2x^2\bigg].\) </p>
</li>
</ul>

  </div>
</div> </p>
<p>In what follows, let \((q_{n})\) be a sequence in \((0,1)\) such that </p>
<div class="displaymath" id="a0000000009">
  \begin{align*}  \lim _{n\rightarrow \infty }q_n& =1,\, \, \, \, \lim _{n\rightarrow \infty }q_n^n=c,\, \, \,  0\leq c{\lt}1,\, \\ \lim _{n\rightarrow \infty }a_1(n,q_n)& =p_1\, and\,  \lim _{n\rightarrow \infty }a_0(n,q_n)=p_0. \end{align*}
</div>
<p><div class="remark_thmwrapper " id="r2">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.3</span>
  </div>
  <div class="remark_thmcontent">
  <p> From <a href="#l2">lemma 2.2</a>, after simple calculations, one has </p>
<div class="displaymath" id="a0000000010">
  \begin{align*} & (i) \lim _{n\rightarrow \infty }[n]_{q_{n}}\mathcal{G}^{M,1}_{n,\alpha }(\phi ^{1}_{x};q_n;x)=\nonumber \\ & =(1-c)\bigg[x^2(p_1(2\alpha -1)-4p_0(1-\alpha ))+x((1-2\alpha )+p_0(1+\alpha ))\bigg] \\ & \quad +(p_0+p_1)(1-2x),\\ & (ii)\lim _{n\rightarrow \infty }[n]_{q_{n}}\mathcal{G}^{M,1}_{n,\alpha }(\phi ^{2}_{x};q_n;x)= \\ & = p_1\Big\{ 4(1-c)x^3(1-\alpha )(1-x)+6x^4-7x^3\nonumber \\ & \quad \! +\! 6\alpha x^3(1\! -\! x)\! +\! x^2\! +\! x(1\! -\! x)^2\Big\}  \! +\! p_0\Big\{ 6\alpha x^2(1\! -\! x)\nonumber \\ & \quad +6x^3-8x^2+2x+4(1-c)x^2(1-\alpha )(1-x)\Big\} . \end{align*}
</div>
<p> Similarly, by carrying out further calculations it can be seen that </p>
<div class="displaymath" id="a0000000011">
  \[ \mathcal{G}^{M,1}_{n,\alpha }(\phi ^{4}_{x};q_n;x)={\mathcal O}\left( [n]_{q_{n}}^{-2}\right), \, \, as\, \,  n\to \infty .  \]
</div>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="bt1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.4</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(\zeta \in \mathcal{C}(\mathcal{I})\) and \(x\in \mathcal{I}\), we have \(|\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)|\leq \| {\zeta }\| .\) </p>

  </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> Using <a href="#l1">lemma 2.1</a>, for every \(x\in \mathcal{I}\) we have, </p>
<div class="displaymath" id="a0000000013">
  \begin{align*} \label{bt1} |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)|& =\sum ^n_{k=0}p^{M,1}_{n,k,\alpha }(q;x)\Big|\zeta \left(\tfrac {[k]_{q_n}}{[n]_{q_n}}\right)\Big| \leq \Vert {\zeta }\Vert \mathcal{G}^{M,1}_{n,\alpha }(1,q_n;x)=\| \zeta \| . \end{align*}
</div>
<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> </p>
<h1 id="a0000000015">3 Rate of convergence of the operators \(\mathcal{G}^{M,1}_{\lowercase {n},\lowercase {\alpha }}\left(.,\lowercase {q_n};\lowercase {x}\right)\)</h1>
<p>First, we prove the uniform convergence theorem for the operators \((\ref{je2})\). </p>
<p><div class="theorem_thmwrapper " id="dkt1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\zeta \in \mathcal{C}(\mathcal{I})\). Then \(\lim _{n\rightarrow \infty } \mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)=\zeta (x)\), uniformly in \(x\in \mathcal{I}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000016">
  <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="#l1">lemma 2.1</a>, it follows that \(\mathcal{G}^{M,1}_{n,\alpha }(e_{i},q;x)\rightarrow e_{i}(x),~ \mbox{as n}\rightarrow \infty \), uniformly in \( \mathcal{I}\), for \(i=0,1,2\). Hence applying Bohman-Korovkin Theorem <span class="cite">
	[
	<a href="#GD" >21</a>
	]
</span>, we obtain the required result. <div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Next, we establish a Voronovskaja type asymptotic result for the operators \((\ref{je2})\). </p>
<p><div class="theorem_thmwrapper " id="dkt2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(\zeta \in \mathcal{C}^2(\mathcal{I})\), then </p>
<div class="displaymath" id="a0000000018">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_{n}}[\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)]= \\ & =\bigg\{ (1-c)\bigg(x^2(p_1(2\alpha -1)-4p_0(1-\alpha ))+x((1-2\alpha )+p_0(1+\alpha ))\bigg)\\ & \quad +(p_0+p_1)(1-2x)\bigg\} \zeta ^{\prime }(x)+\bigg[ p_1\bigg\{ 4(1-c)x^3(1-\alpha )(1-x)+6x^4-7x^3\\ & \quad \! +\! 6\alpha x^3(1-x)\! +\! x^2\! +\! x(1-x)^2\bigg\} +p_0\bigg\{ 4(1-c)x^2(1-\alpha )(1-x)+6x^3-8x^2\\ & \quad +6\alpha x^2(1-x)+2x\bigg\} \bigg]\tfrac {\zeta ^{\prime \prime }(x)}{2}\nonumber , \end{align*}
</div>
<p> uniformly in \(x\in \mathcal{I}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By the Taylor’s expansion of \(\zeta \) about the point \(\hslash =x\), we have </p>
<div class="equation" id="de1">
<p>
  <div class="equation_content">
    \begin{equation} \label{de1} \zeta (\hslash )=\zeta (x)+\zeta ^{\prime }(x)(\hslash -x)+\tfrac {1}{2}\zeta ^{\prime \prime }(x)(\hslash -x)^{2}+\tfrac {1}{2}(\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ,\nonumber \end{equation}
  </div>
  <span class="equation_label">3.2</span>
</p>
</div>
<p> where \(\theta \) lies between \(\hslash \) and \(x\). Operating by \(\mathcal{G}^{M,1}_{n,\alpha }(.,q_n;x)\) in the above equation, we obtain </p>
<div class="displaymath" id="de2">
  \begin{align} \label{de2} \mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)=& \mathcal{G}^{M,1}_{n,\alpha }((\hslash \! -\! x);q_n;x)\zeta ^{\prime }(x)\! +\! \mathcal{G}^{M,1}_{n,\alpha }\left((\hslash \! -\! x)^{2};q_n;x\right)\tfrac {1}{2}\zeta ^{\prime \prime }(x)\nonumber \\ & \quad +\tfrac {1}{2}\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right). \end{align}
</div>
<p>Using the well known properties of modulus of continuity, for any \(\rho {\gt}0\), we have </p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  |\zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)|\leq \omega \left(\zeta ^{\prime \prime };|\theta -x|\right) \leq \left(1+\tfrac {|\hslash -x|}{\rho }\right)\omega \left(\zeta ^{\prime \prime };\rho \right). \end{equation*}
</div>
<p>Therefore, </p>
<div class="displaymath" id="a0000000021">
  \begin{align*} & \Big|\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right)\Big|\leq \\ & \leq \mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}|\zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)|;q_n;x\right)\\ & \leq \omega (\zeta ^{\prime \prime };\rho )\mathcal{G}^{M,1}_{n,\alpha }\left(((\hslash -x)^{2}+\tfrac {1}{\rho }|\hslash -x|^{3});q_n;x\right), \rho {\gt}0. \end{align*}
</div>
<p>Using Cauchy-Schwarz inequality, <a href="#r2">remark 2.3</a> and choosing \(\rho =[n]_{q_{n}}^{-\frac{1}{2}}\), we get </p>
<div class="displaymath" id="a0000000022">
  \begin{align*} & \Big|\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right)\Big|\leq \\ & \leq \omega (\zeta ^{\prime \prime };\rho )\bigg[ \mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2};q_n;x\right) \\ & \quad +\tfrac {1}{\rho }\sqrt{\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^{2};q_n;x)}\sqrt{\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^{4};q_n;x)}\bigg] \\ & =\omega (\zeta ^{\prime \prime };\rho )\bigg[{\mathcal O}\left(\tfrac {1}{[n]_{q_n}}\right)+\tfrac {1}{\rho }{\mathcal O}\left(\tfrac {1}{[n]_{q_n}^{\frac{1}{2}}}\right){\mathcal O}\left(\tfrac {1}{[n]_{q_n}}\right)\bigg] \\ & =\omega (\zeta ^{\prime \prime };[n]^{-\tfrac {1}{2}}_{q_n}){\mathcal O}\left(\tfrac {1}{[n]_{q_n}}\right), \end{align*}
</div>
<p> as \(n\to \infty \), uniformly in \(x \in \mathcal{I}\). </p>
<p>Hence, </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  [n]_{q_n}\Big|\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right)\Big|=\omega \left(\zeta ^{\prime \prime };[n]_{q_n}^{-\frac{1}{2}}\right){\mathcal O}(1), \end{equation*}
</div>
<p> as \(n\to \infty \), uniformly in \(x \in \mathcal{I}\). </p>
<p>Consequently, </p>
<div class="equation" id="a0000000024">
<p>
  <div class="equation_content">
    \begin{equation}  \lim _{n\rightarrow \infty }[n]_{q_n}\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right)=0\nonumber , \end{equation}
  </div>
  <span class="equation_label">3.3</span>
</p>
</div>
<p> uniformly in \(x\in \mathcal{I}\). Thus, from equation \(\eqref{de2}\) and <a href="#r2">remark 2.3</a>, we get </p>
<div class="displaymath" id="a0000000025">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\left[\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)\right]=\\ & = \lim _{n\rightarrow \infty }[n]_{q_n}\bigg\{ \mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)\zeta ^{\prime }(x) + \mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2},q_n;x\right) \tfrac {1}{2}\zeta ^{\prime \prime }(x)\nonumber \\ &  \quad + \tfrac {1}{2}\mathcal{G}^{M,1}_{n,\alpha }\left((\hslash -x)^{2}\{ \zeta ^{\prime \prime }(\theta )-\zeta ^{\prime \prime }(x)\} ;q_n;x\right)\bigg\} \nonumber \\ & =\bigg\{ (1-c)\bigg(x^2(p_1(2\alpha -1)-4p_0(1-\alpha ))+x((1-2\alpha )+p_0(1+\alpha ))\bigg) \\ & \quad +(p_0+p_1)(1-2x)\bigg\} \zeta ^{\prime }(x)+ \bigg[p_1\bigg\{ 4(1-c)x^3(1-\alpha )(1-x)+6x^4-7x^3 \\ &  \quad +6\alpha x^3(1\! -\! x)\! +\! x^2\! +\! x(1-x)^2\bigg\} +p_0\bigg\{ 4(1-c)x^2(1-\alpha )(1-x)+6x^3-8x^2 \\ & \quad +6\alpha x^2(1-x)+2x\bigg\} \bigg]\tfrac {\zeta ^{\prime \prime }(x)}{2}\nonumber , \end{align*}
</div>
<p> uniformly in \(x \in \mathcal{I}\), which completes the proof. <div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, we obtain an approximation theorem for the operators \((\ref{je2})\) by virtue of the second order modulus of continuity by using a smoothing process, <i class="it">e.g.</i> Steklov mean. For \(\zeta \in \mathcal{C}(\mathcal{I})\), the Steklov mean is defined as </p>
<div class="equation" id="le1">
<p>
  <div class="equation_content">
    \begin{equation} \label{le1} \zeta _{h}(x)=\tfrac {4}{h^{2}}\int _{0}^{\tfrac {h}{2}}\int _{0}^{\frac{h}{2}}[2\zeta (x+u+v)-\zeta (x+2(u+v))]dudv. \end{equation}
  </div>
  <span class="equation_label">3.0</span>
</p>
</div>
<p> It is known that for the function \(\zeta _{h}(x)\), there hold the following properties: </p>
<ul class="itemize">
  <li><p>\(\| \zeta _{h}-\zeta \| \leq \omega _{2}(\zeta ,h)\) </p>
</li>
  <li><p>\(\zeta ^{\prime }_{h}, \zeta ^{\prime \prime }_{h}\in \mathcal{C}(\mathcal{I})\quad and\quad \| \zeta ^{\prime }_{h}\| \leq \tfrac {5}{h}\omega (\zeta ,h), ~  \| \zeta ^{\prime \prime }_{h}\| \leq \tfrac {9}{h^{2}}\omega _{2}(\zeta ,h)\), </p>
</li>
</ul>
<p> where \(\omega (\zeta ;h)\) and \(\omega _{2}(\zeta ;h)\) denote the first and second order modulus of continuity. In what follows, for all \(x\in \mathcal{I}\), let \(\mu ^{\alpha ,m}_{n,q_n}(x):=\mathcal{G}^{M,1}_{n,\alpha }\left(\phi ^{m}_{x}(\hslash );q_n;x\right), m\in \mathbb {N}_0:=\mathbb {N}\cup \{ 0\} \) and \(\nu ^{\alpha ,m}_{n, q_n}:=\sup _{x\in \mathcal{I}}\left|\mu ^{\alpha ,m}_{n,q_n}(x)\right|\), \(m\in \mathbb {N}\). </p>
<p><div class="theorem_thmwrapper " id="a0000000027">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\zeta \in \mathcal{C}(\mathcal{I})\). Then following inequality holds: </p>
<div class="equation" id="a0000000028">
<p>
  <div class="equation_content">
    \begin{equation}  \| \mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n.)-\zeta \| \leq 5\omega \left(\zeta ;\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\right)+\tfrac {13}{2}\omega _{2}\left(\zeta ;\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\right)\nonumber . \end{equation}
  </div>
  <span class="equation_label">3.1</span>
</p>
</div>

  </div>
</div> </p>
<p><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> Using the Steklov mean \(\zeta _{h}(x)\) given by \(\eqref{le1}\), we may write </p>
<div class="displaymath" id="le2">
  \begin{align} \label{le2} |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|& \leq \mathcal{G}^{M,1}_{n,\alpha } (|\zeta -\zeta _{h}|;q_n;x+ \\ \nonumber & \quad +|\mathcal{G}^{M,1}_{n,\alpha }(\zeta _{h};q_n;x)-\zeta _{h}(x)|+|\zeta _{h}(x)-\zeta (x)|. \end{align}
</div>
<p> Hence using <a href="#bt1">lemma 2.4</a> and property (a) of Steklov mean, we have </p>
<div class="equation" id="le3">
<p>
  <div class="equation_content">
    \begin{equation} \label{le3} \mathcal{G}^{M,1}_{n,\alpha }(|\zeta -\zeta _{h}|;q_n;x)\leq \| \zeta -\zeta _{h}\| \leq \omega _{2}(\zeta ,h). \end{equation}
  </div>
  <span class="equation_label">3.2</span>
</p>
</div>
<p> Since \(\zeta ^{\prime \prime }_{h} \in \mathcal{C}(\mathcal{I})\), by Taylor’s expansion we have </p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  \zeta _{h}(\hslash )=\zeta _{h}(x)+(\hslash -x)\zeta ^{\prime }_{h}(x)+\tfrac {(\hslash -x)^{2}}{2!}\zeta ^{\prime \prime }(\theta ), \end{equation*}
</div>
<p> where \(\theta \) lies between \(\hslash \) and \(x\). Then, applying Cauchy-Schwarz inequality </p>
<div class="equation" id="a0000000031">
<p>
  <div class="equation_content">
    \begin{equation}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta _{h}(\hslash )-\zeta _{h}(x);q_n;x)|\leq \| \zeta ^{\prime }_{h}\| \sqrt{\nu ^{\alpha ,2}_{n,q_n}}+\tfrac {1}{2}\| \zeta ^{\prime \prime }_{h}\| \nu ^{\alpha ,2}_{n,q_n}\nonumber . \end{equation}
  </div>
  <span class="equation_label">3.3</span>
</p>
</div>
<p> Now, applying property (b) of Steklov mean, we obtain </p>
<div class="equation" id="le4">
<p>
  <div class="equation_content">
    \begin{equation} \label{le4} |\mathcal{G}^{M,1}_{n,\alpha }(\zeta _{h}(\hslash )-\zeta _{h}(x);q_n;x)|\leq \tfrac {5}{h}\omega (\zeta ,h)\sqrt{\nu ^{\alpha ,2}_{n,q_n}}+\tfrac {9}{2h^{2}}\omega _{2}(\zeta ,h)\nu ^{\alpha ,2}_{n,q_n}. \end{equation}
  </div>
  <span class="equation_label">3.3</span>
</p>
</div>
<p> Choosing \(h=\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\) and combining the equations <a href="#le2" class="eqref">3.1</a>–<a href="#le4" class="eqref">3.3</a>, we get the desired result. <div class="proof_wrapper" id="a0000000032">
  <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="a0000000033">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any \(\zeta ^{'}\in \mathcal{C}(\mathcal{I})\), we have </p>
<div class="equation" id="a0000000034">
<p>
  <div class="equation_content">
    \begin{equation}  \| \mathcal{G}^{M,1}_{n,\alpha }(\zeta )-\zeta \| \leq |\nu ^{\alpha ,1}_{n,q_n}|\| \zeta ^{\prime }\| +2\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\omega \left(\zeta ^{\prime },\tfrac {1}{2}\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\right)\nonumber . \end{equation}
  </div>
  <span class="equation_label">3.4</span>
</p>
</div>

  </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> Since \(\zeta ^{'}\in \mathcal{C}(\mathcal{I})\), for any \(\hslash ,x\in \mathcal{I}\), we can write </p>
<div class="equation" id="a0000000036">
<p>
  <div class="equation_content">
    \begin{equation}  \zeta (\hslash )-\zeta (x)=\zeta ^{\prime }(x)(\hslash -x)+\int ^{\hslash }_{x}(\zeta ^{\prime }(u)-\zeta ^{\prime }(x))du\nonumber . \end{equation}
  </div>
  <span class="equation_label">3.4</span>
</p>
</div>
<p> Applying the operator \(\mathcal{G}^{M,1}_{n,\alpha }(.;q_n;x)\) on both sides of the above relation, we get </p>
<div class="displaymath" id="a0000000037">
  \begin{align*}  \mathcal{G}^{M,1}_{n,\alpha }(\zeta (\hslash )-\zeta (x);q_n;x) & =\zeta ^{\prime }(x)\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x);q_n;x) \\ & \quad +\mathcal{G}^{M,1}_{n,\alpha }\left(\int ^{\hslash }_{x}(\zeta ^{\prime }(u)-\zeta ^{\prime }(x))du;q_n;x\right)\nonumber . \end{align*}
</div>
<p> Using the well known property of modulus of continuity </p>
<div class="equation" id="a0000000038">
<p>
  <div class="equation_content">
    \begin{equation}  |\zeta ^{\prime }(u)-\zeta ^{\prime }(x)|\leq \omega (\zeta ^{\prime },\rho )\left(\tfrac {|u-x|}{\rho }+1\right), \rho >0\nonumber , \end{equation}
  </div>
  <span class="equation_label">3.3</span>
</p>
</div>
<p> we obtain </p>
<div class="displaymath" id="a0000000039">
  \begin{align*}  \left|\int ^{\hslash }_{x}(\zeta ^{\prime }(u)-\zeta ^{\prime }(x))du\right|& \leq \left|\int ^{\hslash }_{x}|\zeta ^{\prime }(u)-\zeta ^{\prime }(x)|du\right|\\ & \leq \left|\int ^{\hslash }_{x}\left(1+\tfrac {|u-x|}{\rho }\right)\omega (\zeta ^{\prime },\rho )du\right|\\ & =\omega (\zeta ^{\prime };\rho )\left(|\hslash -x|+\tfrac {(\hslash -x)^{2}}{2\rho }\right)\nonumber . \end{align*}
</div>
<p> Therefore, </p>
<div class="displaymath" id="a0000000040">
  \begin{align*} & |\mathcal{G}^{M,1}_{n,\alpha } (\zeta ;q_n;x)-\zeta (x)|\leq \\ & \leq |\zeta ^{\prime }(x)||\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)|\\ & \quad +\omega (\zeta ^{\prime },\rho )\left\{ \tfrac {1}{2\rho }\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^{2},q_n;x)+\mathcal{G}^{M,1}_{n,\alpha } (|\hslash -x|;q_n;x)\right\} \nonumber . \end{align*}
</div>
<p> Using Cauchy-Schwarz inequality, we have </p>
<div class="displaymath" id="a0000000041">
  \begin{align*} & |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)\! -\! \zeta (x)|\leq \\ \leq & |\zeta ^{\prime }(x)||\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x);q_n;x)| \\ & +\omega (\zeta ^{\prime };\rho )\left\{ \tfrac {1}{2\rho }\sqrt{\mathcal{G}^{M,1}_{n,\alpha } ((\hslash \! -\! x)^{2},q_n;x)}\! +\! 1\right\} \sqrt{\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^{2},q_n;x)}\nonumber . \end{align*}
</div>
<p> Choosing \(\rho =\tfrac {1}{2}\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\), the required assertion is obtained. <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>The following theorem is concerned with an estimate of error in the approximation by the operators (<a href="#je2">1.5</a>) by means of the Peetre’s \(K\)-functional. </p>
<p><div class="theorem_thmwrapper " id="a0000000043">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For all \( \zeta \in \mathcal{C}(\mathcal{I})\) and \( n\in \mathbb {N}\), there exists a constant \( C{\gt}0 \) such that </p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|\leq C \omega _2\bigg(\zeta ;\tfrac {\sqrt{\phi ^{\alpha }_{n,q_n}(x)}}{2}\bigg)+\omega \bigg(\zeta ;\left|\mu _{n,q_n}^{\alpha ,1}(x)\right|\bigg), \end{equation*}
</div>
<p> where \(\phi ^{\alpha }_{n,q_n}(x)=\tfrac {1}{2}\bigg\{ \mu ^{\alpha ,2}_{n,q_n}(x)+(\mu ^{\alpha ,1}_{n,q_n}(x))^{2}\bigg\} \). </p>

  </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> For \(x\in \mathcal{I},\) consider the following auxiliary operator given by </p>
<div class="displaymath" id="je3">
  \begin{eqnarray} \label{je3} \mathcal{G}^{M,1*}_{n,\alpha }(\zeta ;q_n;x)& =\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)+\zeta (x)-\zeta \bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)\bigg). \end{eqnarray}
</div>
<p> It is clear that the operator \(\mathcal{G}^{M,1*}_{n,\alpha }(\zeta ;q_n;x)\) is linear, \(\mathcal{G}^{M,1*}_{n,\alpha }(1;q_n;x)=1\) and </p>
<div class="displaymath" id="a0000000046">
  \begin{align*}  \mathcal{G}^{M,1*}_{n,\alpha }(\hslash -x;q_n;x)& =\mathcal{G}^{M,1}_{n,\alpha }(\hslash -x;q_n;x)-\bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)-x\bigg)=0. \end{align*}
</div>
<p> For every \(\varphi \in \mathcal{C}^2(\mathcal{I})\) and \(\hslash ,x\in \mathcal{I}\), from the Taylor’s theorem, one can write </p>
<div class="displaymath" id="a0000000047">
  \begin{eqnarray*}  \varphi (\hslash )=\varphi (x)+(\hslash -x)\varphi ’(x)+\int _{x}^{\hslash }(\hslash -u)\varphi ”(u)du. \end{eqnarray*}
</div>
<p> Applying \(\mathcal{G}^{M,1*}_{n,\alpha }(.;q;x)\) to the above equation and using <a href="#je3" class="eqref">3.0</a>, we obtain </p>
<div class="displaymath" id="a0000000048">
  \begin{align*}  \mathcal{G}^{M,1*}_{n,\alpha }(\varphi ;q_n;x)=& \varphi (x)+\mathcal{G}^{M,1}_{n,\alpha }\bigg(\int _{x}^{\hslash }(\hslash -u)\varphi ”(u)du;q_n;x\bigg)\\ & -\int _{x}^{\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)}\bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)-u\bigg)\varphi ”(u)du. \end{align*}
</div>
<p>Thus, for all \(x\in \mathcal{I},\) we have </p>
<div class="displaymath" id="chap1eq9">
  \begin{align}  \label{chap1eq9} & |\mathcal{G}^{M,1*}_{n,\alpha }(\varphi ;q;x)-\varphi (x)|\leq \nonumber \\ & \leq \bigg|\mathcal{G}^{M,1}_{n,\alpha }\bigg(\int _{x}^{\hslash }(\hslash -u)\varphi ”(u)du;q_n;x\bigg)\bigg|\nonumber \\ & \quad +\bigg|\int _{x}^{\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)}\bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)-u\bigg)\varphi ”(u)du\bigg|\nonumber \\ & \! \! \leq \! \!  \mathcal{G}^{M,1}_{n,\alpha }\bigg(\bigg|\int _{x}^{\hslash }|\hslash \! \! -\! \! u||\varphi ”(u)|du\bigg|;q_n;x\bigg)\! \! \nonumber \\ & \quad +\! \!  \Bigg|\int _{x}^{\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)}\bigg|\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)\! \! -\! \! u\bigg|\cdot \bigg|\varphi ”(u)\bigg|du\Bigg|\nonumber \\ & \leq \tfrac {\| \varphi ”\| }{2}\bigg[\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2;q;x)+\bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)-x\bigg)^2\bigg]=\| \varphi ”\| \phi ^{\alpha }_{n,q_n}(x). \end{align}
</div>
<p>Further, for all \(x\in \mathcal{I},\) in view of <a href="#bt1">lemma 2.4</a>, we get </p>
<div class="displaymath" id="chap1eq10">
  \begin{align} \label{chap1eq10} |\mathcal{G}^{M,1*}_{n,\alpha }(\zeta ;q_n;x)|\leq |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)|+|\zeta (x)| & +\bigg|\zeta \bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)\bigg)\bigg|\leq 3\| \zeta \| . \end{align}
</div>
<p> Now, for \(\zeta \in \mathcal{C}(\mathcal{I})\) and any \( \varphi \in \mathcal{C}^2(\mathcal{I}) \), using <a href="#je3" class="eqref">3.0</a>–<a href="#chap1eq10" class="eqref">3.2</a>, we obtain </p>
<div class="displaymath" id="a0000000049">
  \begin{align*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|& \leq |\mathcal{G}^{M,1*}_{n,\alpha }(\zeta -\varphi ;q_n;x)-(\zeta -\varphi )(x)|\nonumber \\ & \quad \! +\! \bigg|\zeta \bigg(\mathcal{G}^{M,1}_{n,\alpha }(\hslash ;q_n;x)\bigg)-\zeta (x)\bigg|\! +\! |\mathcal{G}^{M,1*}_{n,\alpha }(\varphi ;q_n;x)-\varphi (x)| \\ & \leq 4\left(\| \zeta -\varphi \| +\tfrac {1}{4}\phi ^{\alpha }_{n,q_n}(x)\| \varphi ”\| \right)+\omega \bigg(\zeta ; |\mu ^{\alpha ,1}_{n,q_n}(x)|\bigg). \end{align*}
</div>
<p> Now, taking the infimum on the right hand side over all \(\varphi \in \mathcal{C}^2(\mathcal{I})\), </p>
<div class="displaymath" id="a0000000050">
  \begin{equation*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|\leq 4K_2\bigg(\zeta ;\tfrac {\phi ^{\alpha }_{n,q_n}(x)}{4}\bigg)+\omega \bigg(\zeta ;|\mu ^{\alpha ,1}_{n,q_n}(x)|\bigg), \end{equation*}
</div>
<p> where \(K_2(\zeta ,\rho )=\inf \left\{ \left\Vert \zeta -\varphi \right\Vert + \rho \left\Vert \varphi ^{\prime \prime }\right\Vert :\varphi \in \mathcal{C}^2(\mathcal{I})\right\} .\)<br />Finally, using the relation between \(K\)-functional and second order modulus of continuity <span class="cite">
	[
	<a href="#DL" >14</a>
	]
</span> given by </p>
<div class="displaymath" id="a0000000051">
  \[ K_2(\zeta ,\rho )\le C^{\prime } \omega _2(\zeta ;\sqrt{\rho }), \]
</div>
<p> \(C^{\prime }\) being some constant, we get </p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|\leq C\omega _2\bigg(\zeta ;\tfrac {\sqrt{\phi ^{\alpha }_{n,q_n}(x)}}{2}\bigg)+\omega \bigg(\zeta ;|\mu ^{\alpha ,1}_{n,q_n}(x)|\bigg), \end{equation*}
</div>
<p> This completes the proof. <div class="proof_wrapper" id="a0000000053">
  <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 our next result, we discuss a convergence estimate by the operators (<a href="#je2">1.5</a>) for the continuous functions on \(\mathcal{I}\) belonging to the Lipschitz class. Let </p>
<div class="displaymath" id="a0000000054">
  \[  \operatorname {Lip}_\theta M=\left\{ \zeta \in \mathcal{C}(\mathcal{I}): |\zeta (\hslash )-\zeta (x)|\leq M|\hslash -x|^\theta , \forall \hslash ,x\in \mathcal{I},0{\lt}\theta \leq 1, M{\gt}0\right\}  \]
</div>
<p> be the Lipschitz class of continuous functions. <div class="theorem_thmwrapper " id="a0000000055">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\zeta \in \operatorname {Lip}_\theta M\). Then for all \(x\in \mathcal{I},\) </p>
<div class="displaymath" id="a0000000056">
  \begin{equation*}  \Big|\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)\Big|\leq M\Big(\mu ^{\alpha ,2}_{n,q_n}(x)\Big)^{\frac{\theta }{2}}. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000057">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By the definition of Lipschitz class, we have </p>
<div class="displaymath" id="a0000000058">
  \begin{align*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)|& \leq \mathcal{G}^{M,1}_{n,\alpha }(|\zeta (\hslash )-\zeta (x)|;q_n;x)\\ & \leq \sum ^{n}_{k=0}P^{M,1}_{n,k,\alpha }(q_n;x)\left|\zeta \left(\tfrac {[k]_{q_n}}{[n]_{q_n}}\right)-\zeta (x)\right|\\ & \leq M\sum ^{n}_{k=0}P^{M,1}_{n,k,\alpha }(q_n;x)\bigg|\tfrac {[k]_{q_n}}{[n]_{q_n}}-x\bigg|^{\theta }. \end{align*}
</div>
<p>Applying Hölder’s inequality, we get </p>
<div class="displaymath" id="a0000000059">
  \begin{align*}  |\mathcal{G}^{M,1}_{n,\alpha }(\zeta ;q_n;x)-\zeta (x)|& \leq M\left(\sum ^{n}_{k=0}P^{M,1}_{n,k,\alpha }(q_n;x)\bigg|\tfrac {[k]_{q_n}}{[n]_{q_n}}-x\bigg|^{2}\right)^{\frac{\theta }{2}}\\ & = M\{ \mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^{2};q_n;x)\} ^{\frac{\theta }{2}} = M\left(\mu ^{\alpha ,2}_{n,q_n}(x)\right)^{\frac{\theta }{2}}, \end{align*}
</div>
<p> which proves the required result. <div class="proof_wrapper" id="a0000000060">
  <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>Finally, we investigate Grüss-Voronovskaya type theorem for the operators (<a href="#je2">1.5</a>). </p>
<p><div class="theorem_thmwrapper " id="a0000000061">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For \( \zeta ^{\prime \prime }, \nu ^{\prime \prime } \in \mathcal{C}(\mathcal{I})\), there holds the following equality: </p>
<div class="displaymath" id="a0000000062">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta \nu ,q_n;x)-\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\mathcal{G}^{M,1}_{n,\alpha }(\nu ,q_n;x)\} = \\ & \! =\! \left(p_1\{ 4(1\! -\! c)x^3(1\! -\! \alpha )(1-x)\! +\! 6x^4-7x^3+6\alpha x^3(1-x)+x^2+x(1-x)^2\} \right.\\ & \quad \! +\! \left.p_0\{ 4(1-c)x^2(1-\alpha )(1-x)\! +\! 6x^3-8x^2+6\alpha x^2(1-x)\! +\! 2x\} \right)\zeta ^\prime (x) \nu ^\prime (x), \end{align*}
</div>
<p> uniformly in \(x\in \mathcal{I}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000063">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For the operators \(\mathcal{G}^{M,1}_{n,\alpha }(.,q_n;x)\), by our hypothesis we may write </p>
<div class="displaymath" id="a0000000064">
  \begin{align*} & [n]_{q_n}\bigg\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta \nu ,q_n;x)-\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\mathcal{G}^{M,1}_{n,\alpha }(\nu ,q_n;x)\bigg\} = \\ & =[n]_{q_n}\bigg\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta \nu ,q_n;x)\\ & -\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)(\zeta \nu )^\prime (x)-\zeta (x) \nu (x)-\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}(\zeta \nu )^{\prime \prime }(x)\\ & \! \! -\! \! \nu (x)\left[\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\! \! -\! \! \zeta (x)-\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)\zeta ^\prime (x)-\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}\zeta ^{\prime \prime }(x)\right]\! \! \\ & \! -\! \mathcal\! \! {G}^{M,1}_{n,\alpha }(\zeta ,q_n;\! x\! )\! \! \left[\mathcal{G}^{M,1}_{n,\alpha }(\nu ,q_n;\! \! x)\! \! -\! \! \nu (x)\! \! \! -\! \! \! \mathcal{G}^{M,1}_{n,\alpha }((\hslash \! \! -\! \! x),q_n\! ;x)\nu ^\prime (x)\! \! -\! \! \tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash \! \! -\! \! x)^2,q_n;x)}{2!}\nu ^{\prime \prime }(x)\right]\\ & \quad +2\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}\zeta ^\prime (x) \nu ^\prime (x) +\nu ^{\prime \prime }(x)\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}\left(\zeta (x)-\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\right) \\ & \quad +\nu ^\prime (x)\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)\left(\zeta (x)-\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\right)\bigg\} . \end{align*}
</div>
<p>Now, in view of <a href="#dkt1">theorem 3.1</a>, for any \(\zeta \in \mathcal{C}(\mathcal{I})\), it follows that \(\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\rightarrow \zeta (x)\), as \(n \rightarrow \infty \), uniformly in \(x\in \mathcal{I}\). Further, following the proof of <a href="#dkt2">theorem 3.2</a>, for any \(\zeta \in \mathcal{C}^2(\mathcal{I})\) we get<br />\([n]_{q_n}\bigg\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)-\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)\zeta ^\prime (x)-\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}\zeta ^{\prime \prime }(x)\bigg\} \rightarrow 0,\,  as\,  n\rightarrow \infty \), uniformly in \(x\in \mathcal{I}\). </p>
<p>Hence, using <a href="#r2">remark 2.3</a>, we get </p>
<div class="displaymath" id="a0000000065">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta \nu ,q_n;x)-\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\mathcal{G}^{M,1}_{n,\alpha }(\nu ,q_n;x)\} =\\ & =\left(p_1\{ 4(1-c)x^3(1-\alpha )(1-x)\! +\! 6x^4\! -\! 7x^3+6\alpha x^3(1-x)+x^2+x(1-x)^2\} \right.\\ & \quad \! +\! \left.p_0\{ 4(1-c)x^2(1-\alpha )(1-x)\! +\! 6x^3-8x^2+6\alpha x^2(1-x)\! +\! 2x\} \right)\zeta ^\prime (x) \nu ^\prime (x), \end{align*}
</div>
<p> which completes the proof. <div class="proof_wrapper" id="a0000000066">
  <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>
<h1 id="a0000000067">4 Bivariate generalization of the operators \(\mathcal{G}^{M,1}_{\lowercase {n},\alpha }(.,\lowercase {q_n};\lowercase {x})\)</h1>
<p>For \(\mathcal{I}^2=\mathcal{I}\times \mathcal{I}\), let \(\mathcal{C}(\mathcal{I}^2)\) be the space of all continuous functions on \(\mathcal{I}^2\), equipped with the norm given by \(\| \zeta \| _{\mathcal{C}(\mathcal{I}^2)}=\sup _{(x_1,x_2)\in \mathcal{I}^2}|\zeta (x_1,x_2)|.\) </p>
<p>For \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and \(0\leq \alpha _1,\alpha _2\leq 1,\) the bivariate generalization of the operator <a href="#je2" class="eqref">1.5</a> is defined by </p>
<div class="displaymath" id="dc4abv1">
  \begin{align} & \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)= \label{dc4abv1} \sum _{k=0}^{n_1}\sum _{j=0}^{n_2} \mathfrak {P}^{\alpha _1,\alpha _2}_{n_1,n_2,k,j}(q_{n_1},q_{n_2};x_1,x_2) \zeta \left(\tfrac {[k]_{q_{n_1}}}{[n_1]_{q_{n_1}}},\tfrac {[j]_{q_{n_2}}}{[n_2]_{q_{n_2}}} \right). \end{align}
</div>
<p> where \(\{ q_{n_i}\} _{n_i\in \mathbb {N}}\) is a sequence in \((0,1)\) satisfying \(\lim \limits _{n_i\rightarrow \infty }q_{n_i}= 1,\,  \lim \limits _{n_i\rightarrow \infty } \, q^{n_i}_{n_i}= c_i,\) \(\lim _{n_i\rightarrow \infty }a_1(n_i,q_{n_i})=\beta _i,\, and\,  \lim _{n_i\rightarrow \infty }a_0(n_i,q_{n_i})=\gamma _i,\; \forall \; i=1,2. \) </p>
<p>Further, \(\mathfrak {P}^{\alpha _1,\alpha _2}_{n_1,n_2,k,j}(q_{n_1},q_{n_2};x_1,x_2)=P^{M,1}_{n_1,k,\alpha _1}(q_{n_1};x_1)P^{M,1}_{n_2,j,\alpha _2}(q_{n_2};x_2),\) where \(P^{M,1}_{n_1,k,\alpha _1}(q_{n_1};x_1)\) and \(P^{M,1}_{n_2,k,\alpha _2}(q_{n_2};x_2)\) are defined similarly as \(P^{M,1}_{n,k,\alpha }(q;x)\) in <a href="#je2" class="eqref">1.5</a>. </p>
<p>Clearly, \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;;x_1,x_2)\) is a linear positive operator. Further, we note that </p>
<div class="displaymath" id="a0000000068">
  \begin{align*} & \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _1-x_1)^u(\hslash _2-x_2)^v;x_1,x_2\right)= \\ & =\mathcal{G}^{q_{n_1}}_{n_1,\alpha _1}\left((\hslash _1-x_1)^u,q_{n_1};x_1\right)\mathcal{G}^{q_{n_2}}_{n_2,\alpha _2}\left((\hslash _2-x_2)^v, q_{n_2};x_2\right)\\ & =\mu ^{\alpha _1,u}_{n_1,q_{n_1}}(x_1)\mu ^{\alpha _2,v}_{n_2,q_{n_2}}(x_2),\forall ~  u,v \in \mathbb {N}_0. \end{align*}
</div>
<p>Let \(e_{pq}(x_1,x_2)=x^{p}_1x^{q}_2,(p,q)\in \mathbb {N}_0\times \mathbb {N}_0\), with \(p+q\leq 2\). In order to establish the results of this section, we require the following Lemmas: </p>
<p><div class="lemma_thmwrapper " id="dc4lbv2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\), we have </p>
<ul class="itemize">
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(1;x_1,x_2)=1;\) </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)=\tfrac {1}{[n_1]_{q_{n_1}}}\bigg[a(x_1,n_1,q_{n_1})\{ (1-\alpha _1)[n_1-3]_{q_{n_1}}q_{n_1}x_1\\ \cdot (1-x_1)+(1-\alpha _1)x_1([2]_{q_{n_1}}+q_{n_1}^3[n-3]_{q_{n_1}}x_1)+\alpha _1[n-1]_{q_{n_1}} q_{n_1}x_1\}  +a(1-x_1,n_1,q_{n_1})\{ (1-\alpha _1)(1-x_1)(1+q_{n_1}^2[n-3]_{q_{n_1}}x_1)+(1-\alpha _1)x_1([3]_{q_{n_1}}+q_{n_1}^4[n-3]_{q_{n_1}}x_1)+\alpha _1(1+q_{n_1}^2[n-1]_{q_{n_1}}x_1)\} \bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{20};x_1,x_2)=\tfrac {a(x_1,n_1,q_{n_1})}{[n]_{q_{n_1}}^2}\bigg[(1-\alpha _1)[n-3]_{q_{n_1}}q_{n_1}x_1(1-x_1)+(1-\alpha _1)[n-3]_{q_{n_1}}[n-4]_{q_{n_1}}q_{n_1}^3x_1^2(1-x_1)+(1-\alpha _1)x_1[2]_{q_{n_1}}^2 +2[2]_{q_{n_1}}(1-\alpha _1)q^3x_1^2[n-3]_{q_{n_1}}+(1-\alpha _1)q^5x_1^2[n-3]_{q_{n_1}}+(1-\alpha _1)q^7x_1^3[n-3]_{q_{n_1}}[n-4]_{q_{n_1}}+\alpha _1[n-1]_{q_{n_1}}q_{n_1}x_1+\alpha _1[n_1-1]_{q_{n_1}}[n_1-2]_{q_{n_1}}q_{n_1}^3x_1\bigg] +\tfrac {a(1-x_1,n_1,q_{n_1})}{[n_1]_{q_{n_1}}^2}\\ \cdot \bigg[(1-\alpha _1)(1-x_1)+2(1-\alpha _1)q_{n_1}^2x_1(1-x_1)[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^3x_1(1-x_1)[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^5x_1^2(1-x_1)[n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}} +(1-\alpha _1)x_1[3]_{q_{n_1}}^2+2(1-\alpha _1)[3]_{q_{n_1}}q_{n_1}^4x_1^2[n_1-3]_{q_{n_1}}+q_{n_1}^7(1-\alpha _1)x_1^2[n_1-3]_{q_{n_1}}+q_{n_1}^9(1-\alpha _1)x_1^3[n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}} +\alpha _1+2q_{n_1}^2\alpha _1 x_1[n_1-1]_{q_{n_1}}+\alpha _1 q_{n_1}^3x_1\\ \cdot [n_1-1]_{q_{n_1}}+\alpha _1 q_{n_1}^5x_1^2[n_1-1]_{q_{n_1}}[n_1-2]_{q_{n_1}}\bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)=\tfrac {1}{[n_2]_{q_{n_2}}}\bigg[a(x_2,n_2,q_{n_2})\{ (1-\alpha _2)[n_2-3]_{q_{n_2}}q_{n_2}x_2\\ \cdot (1-x_2)+(1-\alpha _2)x_2([2]_{q_{n_2}}+q_{n_2}^3[n-3]_{q_{n_2}}x_2)+\alpha _2[n_2-1]_{q_{n_2}} q_{n_2}x_2\}  +a(1-x_2,n_2,q_{n_2})\{ (1-\alpha _2)(1-x_2)(1+q_{n_2}^2[n_2-3]_{q_{n_2}}x_2)+(1-\alpha _2)x_2([3]_{q_{n_2}}+q_{n_2}^4[n-3]_{q_{n_2}}x_2)+\alpha _2(1+q_{n_2}^2[n-1]_{q_{n_2}}x_2)\} \bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{02};x_1,x_2)=\tfrac {a(x_2,n_2,q_{n_2})}{[n_2]_{q_{n_2}}^2}\bigg[(1-\alpha _2)[n_2-3]_{q_{n_2}}q_{n_2}x_2(1-x_2)+(1-\alpha _2)[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}q_{n_2}^3x_2^2(1-x_2)+(1-\alpha _2)x_2[2]_{q_{n_2}}^2 +2[2]_{q_{n_2}}(1-\alpha _2)q_{n_2}^3x_2^2[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^5x_2^2[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^7x_2^3[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}+\alpha _2[n_2-1]_{q_{n_2}}q_{n_2}x_2+\alpha _2[n_2-1]_{q_{n_2}}[n_2-2]_{q_{n_2}}q_{n_2}^3x_2\bigg] +\tfrac {a(1-x_2,n_2,q_{n_2})}{[n_2]_{q_{n_2}}^2}\bigg[(1-\alpha _2)(1-x_2)+2(1-\alpha _2)q_{n_2}^2x_2(1-x_2)[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^3x_2(1-x_2)[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^5x_2^2(1-x_2)[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}} +(1-\alpha _2)x_2[3]_{q_{n_2}}^2+2(1-\alpha _2)[3]_{q_{n_2}}q_{n_2}^4x_2^2[n_2-3]_{q_{n_2}}+q_{n_2}^7(1-\alpha _2)x_2^2[n_2-3]_{}q_{n_2}+q_{n_2}^9(1-\alpha _2)x_2^3[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}} +\alpha _2+2q_{n_2}^2\alpha _2 x_2[n_2-1]_{q_{n_2}}+\alpha _2 q_{n_2}^3x_2[n_2-1]_{q_{n_2}}+\alpha _2 q_{n_2}^5x_2^2[n_2-1]_{q_{n_2}}[n_2-2]_{q_{n_2}}\bigg]\). </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="dc4l3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p> From <a href="#dc4lbv2">lemma 4.1</a>, by a simple calculation we have </p>
<ul class="itemize">
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)=\tfrac {1}{[n_1]_{q_{n_1}}}\bigg[a(x_1,n_1,{q_{n_1}})\{ (1-\alpha _1)[n_1-3]_{q_{n_1}}\\ \cdot q_{n_1}x_1(1-x_1)+(1-\alpha _1)x_1([2]_{q_{n_1}}+q_{n_1}^3[n_1-3]_{q_{n_1}}x_1)+\alpha _1[n_1-1]_{q_{n_1}} q_{n_1}x_1\}  +a(1-x_1,n_1,q_{n_1})\{ (1-\alpha _1)(1-x_1)(1+q_{n_1}^2[n_1-3]_{q_{n_1}}x_1)+(1-\alpha _1)x_1([3]_{q_{n_1}}+q_{n_1}^4[n_1-3]_{q_{n_1}}x_1)+\alpha _1(1+q_{n_1}^2[n_1-1]_{q_{n_1}}x_1)\} -[n_1]_{q_{n_1}}x_1\bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1;x_2)=\tfrac {a(x_1,n_1,q_{n_1})}{[n_1]_{q_{n_1}}^2}\bigg[(1-\alpha _1)[n_1-3]_{q_{n_1}}2x_1(1-x_1)+(1-\alpha _1)[n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}}q_{n_1}^3x_1^2(1-x_1)+(1-\alpha _1)x_1[2]_{q_{n_1}}^2+2[2]_{q_{n_1}}(1-\alpha _1)q_{n_1}^3x_1^2[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^5x_1^2[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^7x_1^3\\ \cdot [n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}}+\alpha _1[n_1-1]_{q_{n_1}}q_{n_1}x_1+\alpha _1[n_1-1]_{q_{n_1}}q_{n_1}x_1+\alpha _1[n_1-1]_{q_{n_1}}[n_1-2]_{q_{n_1}}q_{n_1}^3x_1^2-2x_1^2(1-x_1)q_{n_1}(1-\alpha _1)[n_1-3]_{q_{n_1}}[n_1]_{q_{n_1}}-2(1-\alpha _1)x_1^2[n_1]_{q_{n_1}}([2]_{q_{n_1}}+q_{n_1}^3[n_1-3]_{q_{n_1}}x_1)-2\alpha _1[n_1]_{q_{n_1}}[n_1-1]_{q_{n_1}}q_{n_1}x_1^2+[n_1]_{q_{n_1}}^2x_1^2\bigg] +\tfrac {a(1-x_1,n_1,q_{n_1})}{[n_1]^2}\bigg[(1-\alpha _1)(1-x_1)+2(1-\alpha _1)q_{n_1}^2x_1(1-x_1)[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^3x_1(1-x_1)[n_1-3]_{q_{n_1}}+(1-\alpha _1)q_{n_1}^5x_{1}^2(1-x_1)[n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}}+(1-\alpha _1)x_1[3]_{q_{n_1}}^2+2(1-\alpha _1)[3]_{q_{n_1}}q_{n_1}^4x_1{q_{n_1}}^2 [n_1-3]_{q_{n_1}}+q_{n_1}^7(1-\alpha _1)x_{1}^2[n_1-3]_{q_{n_1}}+q_{n_1}^9(1-\alpha _1) x_{1}^3[n_1-3]_{q_{n_1}}[n_1-4]_{q_{n_1}}+\alpha _1+2q_{n_1}^2\alpha _1 x_1[n_1-1]_{q_{n_1}}+\alpha _1 q_{n_1}^3x_1[n_1-1]_{q_{n_1}}+\alpha _1 q_{n_1}^5x_{1}^2[n_1-1]_{q_{n_1}}[n_1-2]_{q_{n_1}}-2x_1[n_1]_{q_{n_1}}(1-\alpha _1)(1-x_1)(1+q_{n_1}^2[n_1-3]_{q_{n_1}}x_1)-2x_{1}^2(1-\alpha _1)([3]_{q_{n_1}}+q_{n_1}^4[n_1-3]_{q_{n_1}}x_1)[n_1]_{}q_{n_1}-2x_1\alpha _1(1+q_{n_1}^2[n_1-1]_{q_{n_1}}x_1)[n_1]_{q_{n_1}}+[n_1]_{q_{n_1}}^2x_{1}^2\bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)=\tfrac {1}{[n_2]_{q_{n_2}}}\bigg[a(x_2,n_2,{q_{n_2}})\{ (1-\alpha _2)[n_2-3]_{q_{n_2}}\\ \cdot q_{n_2}x_2(1-x_2)+(1-\alpha _2)x_2([2]_{q_{n_2}}+q_{n_2}^3[n_2-3]_{q_{n_2}}x_2)+\alpha _2[n_2-1]_{q_{n_2}} q_{n_2}x_2\}  +a(1-x_2,n_2,q_{n_2})\{ (1-\alpha _2)(1-x_2)(1+q_{n_2}^2[n_2-3]_{q_{n_2}}x_2)+(1-\alpha _2)x_2([3]_{q_{n_2}}+q_{n_2}^4[n_2-3]_{q_{n_2}}x_2)+\alpha _2(1+q_{n_2}^2[n_2-1]_{q_{n_2}}x_2)\} -[n_2]_{q_{n_2}}x_2\bigg]\); </p>
</li>
  <li><p>\(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1;x_2)=\tfrac {a(x_2,n_2,q_{n_2})}{[n_2]_{q_{n_2}}^2}\bigg[(1-\alpha _2)[n_2-3]_{q_{n_2}}2x_2(1-x_2)+(1-\alpha _2)[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}q_{n_2}^3x_{2}^2(1-x_2)+(1-\alpha _2)x_2[2]_{q_{n_2}}^2+2[2]_{q_{n_2}}(1-\alpha _2)q_{n_2}^3x_{2}^2[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^5x_{2}^2[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^7\\ \cdot x_{2}^3[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}+\alpha _2[n_2-1]_{q_{n_2}}q_{n_2}x_2+\alpha _2[n_2-1]_{q_{n_2}}q_{n_2}x_2+\alpha _2[n_2-1]_{q_{n_2}}[n_2-2]_{q_{n_2}}q_{n_2}^3x_{2}^2-2x_{2}^2(1-x_2)q_{n_2}(1-\alpha _2)[n_2-3]_{q_{n_2}}[n_2]_{q_{n_2}}-2(1-\alpha _2)x_{2}^2[n_2]_{q_{n_2}}([2]_{q_{n_2}}+q_{n_2}^3[n_2-3]_{q_{n_2}}x_2)-2\alpha _2[n_2]_{q_{n_2}}[n_2-1]_{q_{n_2}}q_{n_2}x_{2}^2+[n_2]_{q_{n_2}}^2x_{2}^2\bigg] +\tfrac {a(1-x_2,n_2,q_{n_2})}{[n_2]_{q_{n_2}}^2}\bigg[(1-\alpha _2)(1-x_2)+2(1-\alpha _2)q_{n_2}^2x_2(1-x_2)[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^3x_2(1-x_2)[n_2-3]_{q_{n_2}}+(1-\alpha _2)q_{n_2}^5x_{2}^2(1-x_2)[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}+(1-\alpha _2)x_2[3]_{q_{n_2}}^2+2(1-\alpha _2)[3]_{q_{n_2}}q_{n_2}^4x_2{q_{n_2}}^2[n_2-3]_{q_{n_2}}+q_{n_2}^7(1-\alpha _2)x_{2}^2[n_2-3]_{q_{n_2}}+q_{n_2}^9(1-\alpha _2) x_{2}^3[n_2-3]_{q_{n_2}}[n_2-4]_{q_{n_2}}+\alpha _2+2q_{n_2}^2\alpha _2 x_2[n_2-1]_{q_{n_2}}+\alpha _2 q_{n_2}^3x_2[n_2-1]_{q_{n_2}}+\alpha _2 q_{n_2}^5x_{2}^2[n_2-1]_{q_{n_2}}[n_2-2]_{q_{n_2}}-2x_2[n_2]_{q_{n_2}}(1-\alpha _2)(1-x_2)(1+q_{n_2}^2[n_2-3]_{q_{n_2}}x_2)-2x_{2}^2(1-\alpha _2)([3]_{q_{n_2}}+q_{n_2}^4[n_2-3]_{q_{n_2}}x_2)[n_2]_{q_{n_2}}-2x_2\alpha _2(1+q_{n_2}^2[n_2-1]_{q_{n_2}}x_2)[n_2]_{q_{n_2}}+[n_2]_{q_{n_2}}^2x_{2}^2\bigg]\). </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="dc4l4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4.3</span>
  </div>
  <div class="lemma_thmcontent">
  <p> From <a href="#dc4l3">lemma 4.2</a>, one has<br />\((i)\lim _{n_1\rightarrow \infty }[n_1]_{q_{n_1}}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)=\) </p>
<div class="displaymath" id="a0000000069">
  \begin{align*} & =(1-c_1)\bigg[x_1^2(\beta _1(2\alpha _1-1)-4\gamma _1(1-\alpha _1))+x_1((1-2\alpha _1)+\gamma _1(1+\alpha _1))\bigg]\\ & \quad +(\beta _1+\gamma _1)(1-2x_1); \end{align*}
</div>
<p> \((ii) \lim _{n_2\rightarrow \infty }[n_2]_{q_{n_2}}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)=\) </p>
<div class="displaymath" id="a0000000070">
  \begin{align*} & =(1-c_2)\bigg[x_2^2(\beta _2(2\alpha _2-1)-4\gamma _2(1-\alpha _2))+x_2((1-2\alpha _2)+\gamma _2(1+\alpha _2))\bigg]\\ & \quad +(\beta _2+\gamma _2)(1-2x_2); \end{align*}
</div>
<p> \((iii)\lim _{n_1\rightarrow \infty }[n_1]_{q_{n_1}}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)=\) </p>
<div class="displaymath" id="a0000000071">
  \begin{align*} & = \beta _1\{ 4(1-c_1)x_1^3(1-\alpha _1)(1-x_1)+6x_{1}^4-7x_{1}^3+6\alpha _1 x_{1}^3(1-x_{1})+x_{1}^2\\ & \quad +x_{1}(1-x_{1})^2\} +\gamma _1\{ 4(1-c_1)x_{1}^2(1-\alpha _1)(1-x_{1})\\ & \quad +6x_{1}^3-8x_{1}^2+6\alpha _1 x_{1}^2(1-x_{1})+2x_{1}\} ; \end{align*}
</div>
<p> \((iv)\lim _{n_2\rightarrow \infty }[n_2]_{q_{n_2}}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)=\) </p>
<div class="displaymath" id="a0000000072">
  \begin{align*} & = \beta _2\{ 4(1-c_2)x_{2}^3(1-\alpha _2)(1-x_2)+6x_{2}^4-7x_{2}^3+6\alpha _2 x_{2}^3(1-x_2)+x_{2}^2\\ & \quad +x_2(1-x_2)^2\} +\gamma _2\{ 4(1-c_2)x_2^2(1-\alpha _2)(1-x_2)+6x_{2}^3-8x_{2}^2\\ & \quad +6\alpha _2 x_{2}^2(1-x_2)+2x_2\} . \end{align*}
</div>

  </div>
</div> </p>
<h1 id="a0000000073">5 Convergence estimates for the bivariate operators</h1>
<p>In the following result we show the uniform convergence of \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \), if \(\zeta \in \mathcal{C}(\mathcal{I}^2).\) </p>
<p><div class="theorem_thmwrapper " id="dc4t1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(\zeta \in \mathcal{C}(\mathcal{I}^2),\) then </p>
<div class="displaymath" id="a0000000074">
  \begin{equation*}  \lim _{{n_1,n_2}\to \infty }\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)=\zeta (x_1,x_2), \end{equation*}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2\). </p>

  </div>
</div> </p>
<p><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> From <a href="#dc4lbv2">lemma 4.1</a>, obviously </p>
<div class="displaymath" id="a0000000076">
  \begin{align*} & \lim _{{n_1,n_2}\rightarrow \infty }\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{ij})-e_{ij}\| _{\mathcal{C}(\mathcal{I}^2)}=0, for\,  (i,j)\in \{ (0,0),(1,0),(0,1)\} ,\\ & \lim _{{n_1,n_2}\rightarrow \infty }\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{20}+e_{02})-(e_{20}+e_{02})\| _{\mathcal{C}(\mathcal{I}^2)}=0, \end{align*}
</div>
<p> hence applying the well known theorem given by Volkov <span class="cite">
	[
	<a href="#dc4VIK" >37</a>
	]
</span>, the required result follows. <div class="proof_wrapper" id="a0000000077">
  <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>For \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and \(\rho {\gt}0\), the total modulus of continuity is given by: </p>
<div class="displaymath" id="a0000000078">
  \begin{equation*}  \widetilde{\omega }(\zeta ;\rho )= \sup \{ |\zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)|:\sqrt{(\hslash _1-x_1)^2+(\hslash _2-x_2)^2}\leq \rho \} . \end{equation*}
</div>
<p> Hence, </p>
<div class="displaymath" id="a0000000079">
  \begin{equation*}  |\zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)|\leq \widetilde{\omega }(\zeta ;\sqrt{(\hslash _1-x_1)^2+(\hslash _2-x_2)^2}) \leq \widetilde{\omega }(\zeta ;\rho ), \end{equation*}
</div>
<p> whenever \(\sqrt{(\hslash _1-x_1)^2+(\hslash _2-x_2)^2}\leq \rho , \rho {\gt}0\), and for any \(\lambda {\gt}0\), </p>
<div class="displaymath" id="a0000000080">
  \begin{equation*}  \widetilde{\omega }(\zeta ;\lambda \rho )\leq (1+\lambda )\widetilde{\omega }(\zeta ;\rho ). \end{equation*}
</div>
<p>The partial moduli of continuity with respect to \(x_1\) and \(x_2\) are defined as </p>
<div class="displaymath" id="ts1">
  \begin{align}  \widetilde{\omega _{1}}(\zeta ;\rho )& =\sup \bigg\{ |\zeta (x_{11},x_2)-\zeta (x_{12},x_2)|:x_2\in J\  and\  |x_{11}-x_{12}|\leq \rho , \, \,  \rho {\gt}0\bigg\} , \label{ts1} \\ \widetilde{\omega _{2}}(\zeta ;\rho )& =\sup \bigg\{ |\zeta (x_1,x_{21})-\zeta (x_1,x_{22})|:x_1\in J\  and\  |x_{21}-x_{22}|\leq \rho , \, \,  \rho {\gt}0\bigg\} .\nonumber \label{ts2} \end{align}
</div>
<p> For more details on the moduli of continuity for functions of two variables, we refer to <span class="cite">
	[
	<a href="#AG" >24</a>
	]
</span>. </p>
<p>Let \(\mathcal{C}^2(\mathcal{I}^2)= \{ \zeta \in \mathcal{C}(\mathcal{I}^2):\frac{\partial ^2 \zeta }{\partial x_{1}^i\partial x_{2}^j}\in \mathcal{C}(\mathcal{I}^2), \, 0\leq i+j\leq 2\, ~ i,j\in \mathbb {N}_0\} \) equipped with the norm </p>
<div class="displaymath" id="a0000000081">
  \begin{align*}  ||\zeta || _{C^2(\mathcal{I}^2)}=||\zeta ||_{\mathcal{C}(\mathcal{I}^2)}+\sum _{i=1}^2\left(\left\|  \tfrac {\partial ^i \zeta }{\partial x_{1}^i}\right\| _{\mathcal{C}(\mathcal{I}^2)}+\left\|  \tfrac {\partial ^i \zeta }{\partial x_{2}^i}\right\| _{\mathcal{C}(\mathcal{I}^2)}\right)+\left\| \tfrac {\partial ^2 \zeta }{\partial x_{1}\partial x_{2}}\right\| _{\mathcal{C}(\mathcal{I}^2)}. \end{align*}
</div>
<p> The appropriate Peetre’s \(K\)-functional for the function \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) is given by </p>
<div class="displaymath" id="a0000000082">
  \[ \mathcal{K}(\zeta ;\rho )=\inf _{\tau \in {\mathcal{C}^2(\mathcal{I}^2)}}\{ ||\zeta -\tau ||_{\mathcal{C}(\mathcal{I}^2)}+\rho ||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)}\} , \, \rho {\gt}0. \]
</div>
<p> From <span class="cite">
	[
	<a href="#dc4BB" >11</a>
	, 
	p.
	
	192
	]
</span>, we have </p>
<div class="equation" id="dc4fr1">
<p>
  <div class="equation_content">
    \begin{equation} \label{dc4fr1} \mathcal{K}(\zeta ;\rho )\leq M\{ \overline{\omega _{2}}(\zeta ;\sqrt{\rho })+\min (1,\rho )||\zeta ||_{\mathcal{C}(\mathcal{I}^2)}\} , \rho >0, \end{equation}
  </div>
  <span class="equation_label">5.4</span>
</p>
</div>
<p> where the constant \(M{\gt}0\), is independent of \(\zeta \) and \(\rho \) and \(\overline{\omega _{2}}(\zeta ;\sqrt{\rho })\) is the second order modulus of smoothness. </p>
<p>In our further consideration, let </p>
<div class="displaymath" id="a0000000083">
  \begin{align*}  \sup \limits _{(x_1,x_2)\in \mathcal{I}^2}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)& =\theta ^{q_{n_1}}_{n_1,\alpha _1}, \\ \sup \limits _{(x_1,x_2)\in \mathcal{I}^2}|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)|& =\beta ^{q_{n_1}}_{n_1,\alpha _1}, \\ \sup \limits _{(x_1,x_2)\in \mathcal{I}^2}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2} ((\hslash _2-x_2)^2;x_1,x_2)& =\gamma ^{q_{n_2}}_{n_2,\alpha _2}, \\ \sup \limits _{(x_1,x_2)\in \mathcal{I}^2}|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)|& =\rho ^{q_{n_2}}_{n_2,\alpha _2}. \end{align*}
</div>
<p><div class="theorem_thmwrapper " id="dc4t3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\zeta \in \mathcal{C}(\mathcal{I}^2)\), then we have </p>
<div class="displaymath" id="a0000000084">
  \begin{equation*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)} \leq 2 \widetilde{\omega }(\zeta ; \rho ). \end{equation*}
</div>
<p> where \(\rho =(\theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2})^\frac {1}{2}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000085">
  <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 into account the Cauchy-Schwarz inequality, we may write </p>
<div class="displaymath" id="a0000000086">
  \begin{align*} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)|\leq \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)|;x_1,x_2) \\ & \leq \widetilde{\omega }(\zeta ,\rho )\,  \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((1+\tfrac {\sqrt{(\hslash _1-x_{1})^2+(\hslash _2-x_{2})^2}}{\rho };x_1,x_2\right)\\ & \leq \widetilde{\omega }(\zeta ,\rho ) \left(1\! \! +\! \! \tfrac {1}{\rho }\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1\! \! -\! \! x_{1})^2;x_1,x_2)\! \! +\! \! \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2\! \! -\! \! x_{2})^2;x_1,x_2)}\right). \end{align*}
</div>
<p> Now, choosing \(\rho =(\theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2})^\frac {1}{2}\), the required result is proved. <div class="proof_wrapper" id="a0000000087">
  <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="a0000000088">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For \(\zeta \in \mathcal{C}(\mathcal{I}^2)\), the operator \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) verifies the following inequality: </p>
<div class="displaymath" id="a0000000089">
  \begin{equation*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq 2(\widetilde{\omega _{1}}(\zeta ;\sqrt{\theta ^{q_{n_1}}_{n_1,,\alpha _1}})+\widetilde{\omega _{2}}(\zeta ;\sqrt{\gamma ^{q_{n_2}}_{n_2,\alpha _2}}). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000090">
  <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 of the theorem is a direct consequence of the definitions of the partial moduli of continuity, Cauchy-Schwarz inequality and <a href="#dc4lbv2">lemma 4.1</a>. <div class="proof_wrapper" id="a0000000091">
  <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>Next, we determine the rate of convergence for the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) in terms of Peetre’s \(K\)-functional for any \(\zeta \in \mathcal{C}(\mathcal{I}^2)\). </p>
<p><div class="theorem_thmwrapper " id="t11">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For \(\zeta \in \mathcal{C}(\mathcal{I}^2),\) there holds the following inequality: </p>
<div class="displaymath" id="a0000000092">
  \begin{align*} & \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq \\ & \leq M\bigg\{ \overline{\omega _2}\bigg(\zeta ;\tfrac {1}{2}\sqrt{C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}\bigg) +\min \Big\{ 1,\tfrac {C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}{4}\Big\} \Vert \zeta \Vert _{\mathcal{C}(\mathcal{I}^2)}\bigg\}  \\ & \quad +\widetilde{\omega }\bigg(\zeta ;\sqrt{\left(\beta ^{q_{n_1}}_{n_1,\alpha _1}\right)^2+\left(\rho ^{q_{n_2}}_{n_2,\alpha _2}\right)^2}\bigg), \end{align*}
</div>
<p> where \(C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}=\frac{1}{2}\Big\{ (\sqrt{\theta ^{q_{n_1}}_{n_1,\alpha _1}}+\sqrt{\gamma ^{q_{n_2}}_{n_2,\alpha _2}})^2+(\beta ^{q_{n_1}}_{n_1,,\alpha _1}+\rho ^{q_{n_2}}_{n_2,\alpha _2})^2\Big\} \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000093">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Consider the following auxiliary operator defined as: </p>
<div class="displaymath" id="dc4pk1">
  \begin{align} \label{dc4pk1} \overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\zeta ;x_1,x_2)=& \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)+ \zeta (x_1,x_2)\nonumber \\ & -\zeta \left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2),\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)\right). \end{align}
</div>
<p>Then, in view of <a href="#dc4lbv2">lemma 4.1</a>, we have \(\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(1;x_1,x_2)=1,\) </p>
<div class="displaymath" id="a0000000094">
  \begin{equation*}  \overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}((\hslash _1-x_1);x_1,x_2)=0\,  \,  and\,  \, \overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}((\hslash _2-x_2);x_1,x_2)=0. \end{equation*}
</div>
<p> Let \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\) and \((x_1,x_2)\in \mathcal{I}^2\) be arbitrary. By Taylor’s expansion, we can write </p>
<div class="displaymath" id="dc4mg1">
  \begin{align} \label{dc4mg1} & \tau (\hslash _1,\hslash _2)-\tau (x_1,x_2)=\int _{x_1}^{\hslash _1}(\hslash _1-\phi )\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}d\phi +\tfrac {\partial \tau (x_1,x_2)}{\partial x_2}(\hslash _2-x_2) \nonumber \\ & +\tfrac {\partial \tau (x_1,x_2)}{\partial x_1}(\hslash _1-x_1) +\int _{x_2}^{\hslash _2}(\hslash _2-\psi )\tfrac { \partial ^2 \tau (x_1,\psi )d\psi }{\partial \psi ^2}+\int ^{\hslash _1}_{x_1}\int ^{\hslash _2}_{x_2}\tfrac {\partial ^2 \tau d \phi \, d\psi }{\partial \phi \partial \psi }. \end{align}
</div>
<p>Applying \(\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(.;x_1,x_2)\) on both sides of the equation (<a href="#dc4mg1">5.6</a>) and using (<a href="#dc4pk1">5.5</a>), we obtain </p>
<div class="displaymath" id="a0000000095">
  \begin{align*} & \overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\tau ;x_1,x_2)\! -\! \tau (x_1,x_2)\! = \\ & =\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}\bigg( \int _{x_1}^{\hslash _1}(\hslash _1\! -\! \phi )\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}d\phi ;x_1,x_2\bigg) \\ & \quad +\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}\bigg(\int _{x_2}^{\hslash _2}(\hslash _2-\psi )\tfrac { \partial ^2 \tau (x_1,\psi )}{\partial \psi ^2}d\psi ;x_1,x_2\bigg)\\ & \quad +\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}\left(\int ^{\hslash _1}_{x_1}\int ^{\hslash _2}_{x_2}\tfrac {\partial ^2 \tau }{\partial \phi \partial \psi }d \phi \, d\psi ;x_1,x_2\right)\\ & =\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\bigg(\int _{x_1}^{\hslash _1}(\hslash _1-\phi )\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}d\phi ;x_1,x_2\bigg)\\ & \quad -\int _{x_1}^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)}\bigg(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)-\phi \bigg)\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}d\phi \\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\bigg(\int _{x_2}^{\hslash _2}(\hslash _2-\psi )\tfrac { \partial ^2 \tau (x_1,\psi )}{\partial \psi ^2}d\psi ;x_1,x_2\bigg)\\ & \quad -\int _{x_2}^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)}\bigg(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)-\psi \bigg)\tfrac {\partial ^2 \tau (x_1,\psi )}{\partial \psi ^2}d\psi \\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(\int ^{\hslash _1}_{x_1}\int ^{\hslash _2}_{x_2}\tfrac {\partial ^2 \tau }{\partial \phi \partial \psi }d \phi \, d\psi ;x_1,x_2\right)\\ & \quad +\int ^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)}_{x_1}\int ^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)}_{x_2}\tfrac {\partial ^2 \tau }{\partial \phi \partial \psi }d \phi \, d\psi . \end{align*}
</div>
<p>Hence, applying Cauchy-Schwarz inequality </p>
<div class="displaymath" id="dc4dg1">
  \begin{align} \label{dc4dg1} & \Big|\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\tau ;x_1,x_2)-\tau (x_1,x_2)\Big|\leq \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\bigg(\bigg|\int _{x_1}^{\hslash _1}|\hslash _1-\phi | \bigg|\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}\bigg|d\phi \bigg|;x_1,x_2\bigg)\nonumber \\ & \quad +\bigg|\int _{x_1}^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)}\bigg|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)-\phi \bigg|\bigg|\tfrac {\partial ^2 \tau (\phi ,x_2)}{\partial \phi ^2}\bigg|d\phi \bigg|\nonumber \\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\bigg(\bigg|\int _{x_2}^{\hslash _2}|\hslash _2-\psi |\bigg|\tfrac {\partial ^2 \tau (x_1,\psi )}{\partial \psi ^2}\bigg|d\psi \bigg|;x_1,x_2\bigg)\nonumber \\ & \quad +\bigg|\int _{x_2}^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)}\bigg|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)-\psi \bigg|\bigg|\tfrac {\partial ^2 \tau (x_1,\psi )}{\partial \psi ^2}\bigg|d\psi \bigg|\nonumber \\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(\bigg|\int ^{\hslash _1}_{x_1}\int ^{\hslash _2}_{x_2}\bigg|\tfrac {\partial ^2 \tau }{\partial \phi \partial \psi }\bigg|d \phi \, d\psi \bigg|;x_1,x_2\right)\nonumber \\ & \quad +\bigg|\int ^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)}_{x_1}\int _{x_2}^{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)}\tfrac {\partial ^2 \tau }{\partial \phi \partial \psi }d \phi \, d\psi \bigg|\nonumber \\ & \leq \tfrac {1}{2}\bigg\{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;{x_1,x_2})+\bigg(\mu ^{\alpha _1,1}_{n_1,q_{n_1}}(x_1)\bigg)^2\nonumber \\ & \quad + \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;{x_1,x_2})+\bigg(\mu ^{\alpha _2,1}_{n_2,q_{n_2}}(x_2)\bigg)^2\bigg\} ||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)}\nonumber \\ & \quad +\bigg\{ \sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;{x_1,x_2})}\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;{x_1,x_2})}\nonumber \\ & \quad + \bigg|\left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2)\! \! -\! \! x_1\right)\bigg|\bigg|\left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)\! \! -\! \! x_2\right)\bigg|\bigg\} ||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)}\nonumber \\ & \leq \tfrac {1}{2}\bigg\{ (\sqrt{\theta ^{q_{n_1}}_{n_1,\alpha _1}}\! \! +\! \! \sqrt{\gamma ^{q_{n_2}}_{n_2,\alpha _2}})^2\! \! +\! \! (\beta ^{q_{n_1}}_{n_1,,\alpha _1}\! +\! \rho ^{q_{n_2}}_{n_2,\alpha _2})^2\bigg\} ||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)}\! \! =\! \! C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)}.\nonumber \end{align}
</div>
<p> Also, from <a href="#dc4pk1" class="eqref">5.5</a>, we have </p>
<div class="displaymath" id="dc4mg3">
  \begin{align} \label{dc4mg3} |\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\zeta ;x_1,x_2)|& \leq |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)|+ |\zeta (x_1,x_2)|+\nonumber \\ & \quad \! \! +\! \! \bigg|\zeta \left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2),\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)\right)\bigg|\nonumber \\ & \leq 3\| \zeta \| _{\mathcal{C}(\mathcal{I}^2)}. \end{align}
</div>
<p> Now, for \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and any \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\), using <a href="#dc4dg1" class="eqref">5.7</a> and <a href="#dc4mg3" class="eqref">5.8</a>, we may write </p>
<div class="displaymath" id="a0000000096">
  \begin{align*} & \Big|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)\Big| \leq \\ & \leq |\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\zeta -\tau ;x_1,x_2)| \\ & \quad +|\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(\tau ;x_1,x_2)-\tau (x_1,x_2)|+|\tau (x_1,x_2)-\zeta (x_1,x_2)| \\ & \quad +\bigg|\zeta \left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2), \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)\right)-\zeta (x_1,x_2)\bigg| \\ & {\lt}4\| \zeta -\tau \| _{\mathcal{C}(\mathcal{I}^2)}+C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}||\tau ||_{\mathcal{C}^2(\mathcal{I}^2)} \\ & \quad +\bigg|\zeta \left(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{10};x_1,x_2),\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{01};x_1,x_2)\right)-\zeta (x_1,x_2)\bigg|\\ & \leq 4\bigg(\| \zeta -\tau \| _{\mathcal{C}(\mathcal{I}^2)}+\tfrac {C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}{4}\| \tau \| _{\mathcal{C}^2(\mathcal{I}^2)}\bigg)+\widetilde{\omega }\bigg(\zeta ;\sqrt{\left(\beta ^{q_{n_1}}_{n_1,,\alpha _1}\right)^2+\left(\rho ^{q_{n_2}}_{n_2,\alpha _2}\right)^2}\bigg). \end{align*}
</div>
<p> Taking the infimum over all \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\) on the right hand side of the above equation and using (<a href="#dc4fr1">5.4</a>), we get </p>
<div class="displaymath" id="a0000000097">
  \begin{align*} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)|\leq \\ & \leq 4\mathcal{K}\bigg(\zeta ;\tfrac {C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}{4}\bigg)+\widetilde{\omega }\bigg(\zeta ;\sqrt{\beta ^{2q_{n_1}}_{n_1,,\alpha _1}+\rho ^{2q_{n_2}}_{n_2,\alpha _2}}\bigg)\\ & \leq M\bigg\{ \overline{\omega _2}\bigg(\zeta ;\tfrac {1}{2}\sqrt{C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}\bigg)+\mbox{min}\{ 1,\tfrac {C^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}{4}\} \Vert \zeta \Vert _{\mathcal{C}(\mathcal{I}^2)}\bigg\} \\ & \quad +\widetilde{\omega }\bigg(\zeta ;\sqrt{\left(\beta ^{q_{n_1}}_{n_1,,\alpha _1}\right)^2+\left(\rho ^{q_{n_2}}_{n_2,\alpha _2}\right)^2}\bigg),\,  \forall \,  (x_1,x_2)\in \mathcal{I}^2, \end{align*}
</div>
<p> which leads us to the desired assertion. <div class="proof_wrapper" id="a0000000098">
  <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>The next result provides a convergence estimate for functions in \(\mathcal{C}^{1}(\mathcal{I}^2)= \{ \zeta \in \mathcal{C}(\mathcal{I}^2):\frac{\partial \zeta }{\partial x_{1}},\frac{\partial \zeta }{\partial x_{2}}\in \mathcal{C}(\mathcal{I}^2) \} \) by the operators (<a href="#dc4abv1">4.2</a>). </p>
<p><div class="theorem_thmwrapper " id="a0000000099">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(\zeta \in \mathcal{C}^{1}(\mathcal{I}^2)\), then there holds </p>
<div class="displaymath" id="a0000000100">
  \begin{equation*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq \parallel \zeta ^\prime _{x_1}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\theta ^{q_{n_1}}_{n_1,\alpha _1}}\; +\parallel \zeta ^\prime _{x_2}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\gamma ^{q_{n_2}}_{n_2,\alpha _2}}. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000101">
  <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 \((x_1,x_2)\in \mathcal{I}^2\), be an arbitrary but fixed point. Then, we may write </p>
<div class="displaymath" id="a0000000102">
  \begin{equation*}  \zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)=\int _{x_1}^{\hslash _1} \zeta ^\prime _{\theta _1}(\theta _1,x_2)d\theta _1+\int _{x_2}^{\hslash _2} \zeta ^\prime _{\theta _2}(x_1,\theta _2)d\theta _2. \end{equation*}
</div>
<p>Now, applying \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of the above equation, we get </p>
<div class="displaymath" id="a0000000103">
  \begin{align*} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta (\hslash _1,\hslash _2);x_1,x_2)-\zeta (x_1,x_2)|\leq \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(\left|\int _{x_1}^{\hslash _1} \zeta ^\prime _{\theta _1}(\theta _1,x_2)d \theta _1\right|;x_1,x_2\right)\\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(\left|\int _{x_2}^{\hslash _2} \zeta ^\prime _{\theta _2}(x_1,\theta _2)d\theta _2\right|;x_1,x_2\right). \end{align*}
</div>
<p>By applying the inequalities </p>
<div class="displaymath" id="a0000000104">
  \begin{align*}  \bigg|\int _{x_1}^{\hslash _1} \zeta ^\prime _{\theta _1}(\theta _1,x_2)d\theta _1\bigg| \leq \parallel \zeta ^\prime _{x_1}\parallel _{\mathcal{C}(\mathcal{I}^2)} |\hslash _1-x_1|, \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000105">
  \begin{align*}  \bigg|\int _{x_2}^{\hslash _2}\zeta ^\prime _{\theta _2}(x_1,\theta _2)d\theta _2\bigg| \leq \parallel \zeta ^\prime _{x_2}\parallel _{\mathcal{C}(\mathcal{I}^2)} |\hslash _2-x_2|, \end{align*}
</div>
<p> we get </p>
<div class="displaymath" id="a0000000106">
  \begin{align*} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta (\hslash _1,\hslash _2);x_1,x_2)-\zeta (x_1,x_2)|\leq \\ & \leq \parallel \zeta ^\prime _{x_1}\parallel _{\mathcal{C}(\mathcal{I}^2)} \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1|;x_1,x_2)\\ & \quad +\parallel \zeta ^\prime _{x_2}\parallel _{\mathcal{C}(\mathcal{I}^2)} \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _2-x_2|;x_1,x_2). \end{align*}
</div>
<p> Applying Cauchy-Schwarz inequality and <a href="#dc4l3">lemma 4.2</a>, we obtain </p>
<div class="displaymath" id="a0000000107">
  \begin{align*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}& \leq \parallel \zeta ^\prime _{x_1}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-.)^2)\| _{\mathcal{C}(\mathcal{I}^2)}} \; \; \\ & \quad +\parallel \zeta ^\prime _{x_2}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-.)^2)\| _{\mathcal{C}(\mathcal{I}^2)}}\\ & = \parallel \zeta ^\prime _{x_1}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\theta ^{q_{n_1}}_{n_1,\alpha _1}}\; +\parallel \zeta ^\prime _{x_2}\parallel _{\mathcal{C}(\mathcal{I}^2)} \sqrt{\gamma ^{q_{n_2}}_{n_2,\alpha _2}}. \end{align*}
</div>
<p> This completes the proof. <div class="proof_wrapper" id="a0000000108">
  <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>In the next result, we discuss the convergence behaviour of \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \) by virtue of the partial moduli of continuity of the partial derivatives of \(\zeta \). </p>
<p><div class="theorem_thmwrapper " id="a0000000109">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\zeta \in \mathcal{C}^{1}(\mathcal{I}^2)\), then for sufficiently large \(n_1\) and \(n_2\), there holds the following inequality: </p>
<div class="displaymath" id="a0000000110">
  \begin{equation*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq C\sum ^2_{i=1}[n_i]^{-\frac{1}{2}}_{q_{n_i}}\left(1+2\widetilde{\omega _{i}}\left(\zeta ^{\prime }_{x_i};[n_i]^{-\frac{1}{2}}_{q_{n_i}}\right)\right), \end{equation*}
</div>
<p> where \(\widetilde{\omega _{i}}\left(\zeta ^{\prime }_{x_i};.\right)\) are the partial moduli of continuity of \(\zeta ^{\prime }_{x_i}\) for \(i=1,2\) and \(C\) is some positive constant. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000111">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Using the mean value theorem in the following form, we obtain </p>
<div class="displaymath" id="a0000000112">
  \begin{align*}  \zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)& =(\hslash _1-x_1)\zeta ^{\prime }_{x_1}(u,x_2)+(\hslash _2-x_2)\zeta ^{\prime }_{x_2}(x_1,v)\\ & =(\hslash _1\! -\! x_1)\zeta ^{\prime }_{x_1}(x_1,x_2)\! +\! (\hslash _1\! -\! x_1)\left(\zeta ^{\prime }_{x_1}(u,x_2)-\zeta ^{\prime }_{x_1}(x_1,x_2)\right)\\ & \quad +(\hslash _2\! -\! x_2)\zeta ^{\prime }_{x_2}(x_1,x_2)\! +\! (\hslash _2\! -\! x_2)\left(\zeta ^{\prime }_{x_2}(x_1,v)\! \! -\! \! \zeta ^{\prime }_{x_2}(x_1,x_2)\right) \end{align*}
</div>
<p> where \(u\) and \(v\) lie between \(\hslash _1,x_1\) and \(\hslash _2,x_2\) respectively. Applying \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) \(\left(.;x_1,x_2\right)\) to the above equation, we obtain </p>
<div class="displaymath" id="a0000000113">
  \begin{align*} & \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)=\\ & =\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _1-x_1);x_1,x_2\right)\zeta ^{\prime }_{x_1}(x_1,x_2) \\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _1-x_1)\left(\zeta ^{\prime }_{x_1}(u,x_2)-\zeta ^{\prime }_{x_1}(x_1,x_2)\right);x_1,x_2\right)\\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _2-x_2);x_1,x_2\right)\zeta ^{\prime }_{x_2}(x_1,x_2)\\ & \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _2-x_2)\left(\zeta ^{\prime }_{x_2}(x_1,v)-\zeta ^{\prime }_{x_2}(x_1,x_2)\right);x_1,x_2\right). \end{align*}
</div>
<p>Since \(\zeta ^{\prime }_{x_1}\) and \(\zeta ^{\prime }_{x_1}\) are continuous on \(\mathcal{I}^2\), they are bounded therein, therefore there exist positive constants \(C_1 \) and \(C_2\) such that \(|\zeta ^{\prime }_{x_1}|\leq C_1\) and \(|\zeta ^{\prime }_{x_2}|\leq C_2\), for all \((x_1,x_2)\in \mathcal{I}^2\). Hence applying Cauchy-Schwarz inequality, for any \(\rho _1,\rho _2{\gt}0\), we obtain </p>
<div class="displaymath" id="a0000000114">
  \begin{align*} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)|\leq \sum ^2_{i=1}|\zeta ^{\prime }_{x_i}|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(|\hslash _i-x_i|;x_1,x_2\right) \\ & \quad +\sum ^2_{i=1}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(|\hslash _i-x_i|\left(1+\tfrac {|\hslash _i-x_i|}{\rho _i}\right);x_1,x_2\right)\widetilde{\omega _{i}}(\zeta ^{\prime }_{x_i};\rho _i)\\ & \leq \sum ^2_{i=1}\left(C_{i}\{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _i-x_i)^{2};x_1,x_2\right)\} ^{\frac{1}{2}}\right.\! \! +\! \! \widetilde{\omega _{i}}(\zeta ^{\prime }_{x_i};\rho _i)\\ & \quad \times \bigg[\Big\{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\big((\hslash _i-x_i)^2;x_1,x_2\big)\Big\} ^{\frac{1}{2}} +\tfrac {1}{\rho _i}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left((\hslash _i\! \! -\! \! x_i)^2;x_1,x_2\right)\bigg]\bigg). \end{align*}
</div>
<p> Choosing \(\rho _i=([n_i]_{q_{n_i}})^{-\frac{1}{2}}, i=1,2\) and applying <a href="#dc4l4">lemma 4.3</a>, the required result is proved. <div class="proof_wrapper" id="a0000000115">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, we establish a convergence estimate for the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) with the aid of the Lipschitz class functions. </p>
<p>For \(\zeta :\mathcal{I}^2\rightarrow \mathbb {R}\) and \(0{\lt}\theta \leq 1\), the function \(\zeta \) is said to be in Lipschitz class \(\operatorname {Lip}_{\mathcal{M}}(\theta )\), if \(\exists \) a positive constant \(\mathcal{M}\) such that </p>
<div class="displaymath" id="a0000000116">
  \begin{equation*}  \operatorname {Lip}_{\mathcal{M}}(\theta )=\{ \zeta :|\zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)|\leq \mathcal{M}\| r-x\| ^\theta \} , \end{equation*}
</div>
<p> \(\forall r=(\hslash _1,\hslash _2), x=(x_1,x_2)\in \mathcal{I}^2,\) where \(\| r-x\| =\{ (\hslash _1-x_1)^2+(\hslash _2-x_2)^2\} ^{\frac{1}{2}}\) is the Euclidean norm. </p>
<p><div class="theorem_thmwrapper " id="a0000000117">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\zeta \in \operatorname {Lip}_{\mathcal{M}}(\theta )\), \(0{\lt}\theta \leq 1\). Then for sufficiently large \(n_1\) and \(n_2\), the operators <a href="#dc4abv1" class="eqref">4.2</a> verify the following relation: </p>
<div class="displaymath" id="a0000000118">
  \begin{equation*}  \| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(\zeta \right)-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq \mathcal{K}\{ [n_1]^{-1}_{q_{n_1}}+[n_2]^{-1}_{q_{n_2}}\} ^{\frac{\theta }{2}}, \end{equation*}
</div>
<p> where \(\mathcal{K}\) is some positive constant. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000119">
  <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 hypothesis, we have </p>
<div class="displaymath" id="a0000000120">
  \begin{align*}  |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)| & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\left|\zeta (\hslash _1,\hslash _2)-\zeta (x_1,x_2)\right|;x_1,x_2)\\ & \leq \mathcal{M}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\| r-x\| ^\theta ;x_1,x_2), \end{align*}
</div>
<p> where \(r=(\hslash _1,\hslash _2), x=(x_1,x_2)\in \mathcal{I}^2\). Applying H\(\ddot{o}\)lder’s inequality , we obtain </p>
<div class="displaymath" id="a0000000121">
  \begin{align*}  |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-(\zeta ;x_1,x_2)| & \leq \mathcal{M}\{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\| r-x\| ^2;x_1,x_2)\} ^{\frac{\theta }{2}}\\ & \leq \mathcal{K}\{ \theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2}\} ^{\frac{\theta }{2}},\; \forall \; x_1,x_2\in \mathcal{I}^2, \end{align*}
</div>
<p> hence using <a href="#dc4l3">lemma 4.2</a>, the required assertion is proved. <div class="proof_wrapper" id="a0000000122">
  <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>Next, we discuss a Voronovskaja type asymptotic theorem. </p>
<p><div class="theorem_thmwrapper " id="dc4vo1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.8</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\zeta \in \mathcal{C}^2(\mathcal{I}^2).\) Then </p>
<div class="displaymath" id="a0000000123">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\left(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)\right)= \\ & = \bigg[(1-c)\{ x_1^2(p_1(2\alpha _1-1)-4p_0(1-\alpha _1))+x_1((1-2\alpha _1)+p_0(1+\alpha _1))\} \\ & \quad +(p_0+p_1)(1-2x_1)\bigg]\zeta ^{\prime }_{x_1}(x_1,x_2)+\bigg[(1-c)\{ x_2^2(p_1(2\alpha _2-1)-4p_0(1-\alpha _2))\\ & \quad +x_2((1-2\alpha _2)+p_0(1+\alpha _2))\}  +(p_0+p_1)(1-2x_2)\bigg]\zeta ^{\prime }_{x_2}(x_1,x_2)\\ & \quad +\tfrac {1}{2}\bigg[p_1\{ 4(1-c)x_1^3(1-\alpha _1)(1-x_1)+6x_1^4-7x_1^3+6\alpha _1 x_1^3(1-x_1)+x_1^2\\ & \quad +x_1(1-x_1)^2\} +p_0\{ 4(1-c)x_1^2(1-\alpha _1)(1-x_1)\\ & \quad +6x_1^3-8x_1^2+6\alpha _1 x_1^2(1-x_1)+2x_1\} \bigg]\zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2)\\ & \quad +\tfrac {1}{2}\bigg[p_1\{ 4(1-c)x_2^3(1-\alpha _2)(1-x_2)+6x_2^4-7x_2^3+6\alpha _2 x_2^3(1-x_2)+x_2^2\\ & \quad +x_2(1-x_2)^2\} +p_0\{ 4(1-c)x_2^2(1-\alpha _2)(1-x_2)+6x_2^3-8x_2^2\\ & \quad +6\alpha _2 x_2^2(1-x_2)+2x_2\} \bigg]\zeta ^{\prime \prime }_{x_2x_2}(x_1,x_2), \end{align*}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000124">
  <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 \((x_1,x_2)\in \mathcal{I}^2\) be an arbitrary but fixed point. Using Taylor’s theorem, we have </p>
<div class="displaymath" id="dc4dee1">
  \begin{align} \label{dc4dee1} \zeta (\hslash _1,\hslash _2)& =\zeta (x_1,x_2)+\zeta ^{\prime }_{x_1}(x_1,x_2)(\hslash _1-x_1)+\tfrac {1}{2}\{ \zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2)(\hslash _1-x_1)^2+\nonumber \\ & \quad +2\zeta ^{\prime \prime }_{x_1x_2}(x_1,x_2)(\hslash _1-x_1)(\hslash _2-x_2)+\zeta ^{\prime \prime }_{x_2x_2}(x_1,x_2)(\hslash _2-x_2)^2\} \nonumber \\ & \quad +\varpi (\hslash _1,\hslash _2;x_1,x_2)\sqrt{((\hslash _1\! -\! x_1)^4+(\hslash _2\! -\! x_2)^4)}\! +\! \zeta ^{\prime }_{x_2}(x_1,x_2)(\hslash _2\! -\! x_2), \end{align}
</div>
<p> where \(\varpi (\hslash _1,\hslash _2;x_1,x_2)\in \mathcal{C}(\mathcal{I}^2)\) and \(\varpi (\hslash _1,\hslash _2;x_1,x_2) \rightarrow 0,\) as \((\hslash _1,\hslash _2)\rightarrow (x_1,x_2).\)<br />Operating \(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of (<a href="#dc4dee1">5.9</a>), we have </p>
<div class="displaymath" id="dc4dee2">
  \begin{align} \label{dc4dee2} & \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)=\zeta ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)+ \\ & +\zeta ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)\nonumber \\ & +\tfrac {1}{2}\{ \zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;{x_1,x_2})\nonumber \\ & +2\zeta ^{\prime \prime }_{x_1x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2)\nonumber \\ & +\zeta ^{\prime \prime }_{x_2x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;{x_1,x_2})\} +\zeta (x_1,x_2) \nonumber \\ & +\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\varpi (\hslash _1,\hslash _2;x_1,x_2)\sqrt{((\hslash _1-x_1)^4+(\hslash _2-x_2)^4)};x_1,x_2).\nonumber \end{align}
</div>
<p> Applying Cauchy-Schwarz inequality to the last term of (<a href="#dc4dee2">5.10</a>), we obtain </p>
<div class="displaymath" id="a0000000125">
  \begin{align*} & \bigg|\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}\left(\varpi (\hslash _1,\hslash _2;x_1,x_2)\sqrt{((\hslash _1-x_1)^4+(\hslash _2-x_2)^4)};x_1,x_2\right)\bigg|\leq \\ & \leq \{ \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\varpi ^2(\hslash _1,\hslash _2;x_1,x_2);x_1,x_2)\} ^{1/2}\\ & \quad \times \left\{ \sqrt{\{ \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^4;x_1,x_2)+\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^4;x_1,x_2)\} }\right\} . \end{align*}
</div>
<p>Since \(\varpi (.,.;x_1,x_2)\in \mathcal{C}(\mathcal{I}^2)\) and \(\varpi (\hslash _1,\hslash _2;x_1,x_2)\rightarrow 0,\) as \((\hslash _1,\hslash _2)\rightarrow (x_1,x_2),\) applying <a href="#dc4t1">theorem 5.1</a>, we obtain </p>
<div class="displaymath" id="dc4de3">
  \begin{eqnarray} \label{dc4de3} \displaystyle \lim _{n\rightarrow \infty }[n]_{q_n} \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\varpi (\hslash _1,\hslash _2;x_1,x_2)\sqrt{((\hslash _1-x_1)^4+(\hslash _2-x_2)^4)};x_1,x_2)=0,\nonumber \end{eqnarray}
</div>
<p> \(\mbox{uniformly in} ~ ~ ~ ~ (x_1,x_2)\in \mathcal{I}^2.\) </p>
<p>Now, using <a href="#dc4l4">lemma 4.3</a> and by the above equation, from (<a href="#dc4dee2">5.10</a>) we reach to the required result. <div class="proof_wrapper" id="a0000000126">
  <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>Grüss <span class="cite">
	[
	<a href="#dc4GG" >20</a>
	]
</span> determined the difference between the integral of a product of two functions and the product of integrals of the two functions. Later, Gal and Gonska <span class="cite">
	[
	<a href="#dc4SGH" >18</a>
	]
</span>, studied the Grüss Voronovskaya type theorem for Bernstein and Paltanea operators with the aid of Grüss inequality which deals with the non-multiplicavity of the operators. For more details in this direction, one can see (<span class="cite">
	[
	<a href="#dc4PBR" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#dc4SAT" >28</a>
	]
</span>) and the references therein. In the following theorem, we examine the non-multiplicativity of the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\). </p>
<p><div class="theorem_thmwrapper " id="a0000000127">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.9</span>
    <span class="theorem_thmtitle">Grüss Voronovskaja type theorem</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For \(\zeta , \tau \in \mathcal{C}^{2}(\mathcal{I}^2)\), there holds </p>
<div class="displaymath" id="a0000000128">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\{ \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta \tau ;x_1,x_2)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\tau ;x_1,x_2)\} = \\ & =\bigg[p_1\{ 4(1-c)x_1^3(1-\alpha _1)(1-x_1)+6x_1^4-7x_1^3+6\alpha _1 x_1^3(1-x_1)+x_1^2 \\ & \quad +x_1(1-x_1)^2\} +p_0\{ 4(1-c)x_1^2(1-\alpha _1)(1-x_1) \\ & \quad +6x_1^3-8x_1^2+6\alpha _1 x_1^2(1-x_1)+2x_1\} \bigg]\zeta ^{\prime }_{x_1}(x_1,x_2)\tau ^{\prime }_{x_1}(x_1,x_2) \\ & \quad +\bigg[p_1\{ 4(1-c)x_2^3(1-\alpha _2)(1-x_2)+6x_2^4-7x_2^3+6\alpha _2 x_2^3(1-x_2)+x_2^2\\ & \quad +x_2(1-x_2)^2\} +p_0\{ 4(1-c)x_2^2(1-\alpha _2)(1-x_2)+6x_2^3-8x_2^2\\ & \quad +6\alpha _2 x_2^2(1-x_2)+2x_2\} \bigg]\zeta ^{\prime }_{x_2}(x_1,x_2)\tau ^{\prime }_{x_2}(x_1,x_2), \end{align*}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000129">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By our hypothesis, we obtain </p>
<div class="displaymath" id="a0000000130">
  \begin{align*} & [n]_{q_n}\{ \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta \tau ;x_1,x_2)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\tau ;x_1,x_2)\} = \\ & =[n]_{q_n}\bigg(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta \tau ;x_1,x_2)-\zeta (x_1,x_2)\tau (x_1,x_2)-(\zeta (x_1,x_2) \tau ^{\prime }_{x_1}(x_1,x_2)\\ & \quad +\tau (x_1,x_2) \zeta ^{\prime }_{x_1}(x_1,x_2))\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)-(\zeta (x_1,x_2) \tau _{x_2}(x_1,x_2)\\ & \quad +\tau (x_1,x_2) \zeta ^{\prime }_{x_2}(x_1,x_2))\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)-\tfrac {1}{2}(\zeta (x_1,x_2) \tau ^{\prime \prime }_{x_1x_1}(x_1,x_2)\\ & \quad +2\zeta ^{\prime }_{x_1}(x_1,x_2) \tau ^{\prime }_{x_1}(x_1,x_2)+\tau (x_1,x_2) \zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2))\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)\\ & \quad -(\zeta (x_1,x_2) \tau ^{\prime \prime }_{x_1x_2}(x_1,x_2)+\zeta ^{\prime }_{x_1}(x_1,x_2)\tau ^{\prime }_{x_2}(x_1,x_2)+\zeta ^{\prime }_{x_2}(x_1,x_2)\tau ^{\prime }_{x_1}(x_1,x_2) \\ & \quad +\tau (x_1,x_2) \zeta ^{\prime \prime }_{x_1x_2}(x_1,x_2))\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2) \\ & \quad -\tfrac {1}{2}(\zeta (x_1,x_2)\tau ^{\prime \prime }_{x_2x_2}(x_1,x_2)+2\zeta ^{\prime }_{x_2}(x_1,x_2)\tau ^{\prime }_{x_2}(x_1,x_2) \\ & \quad +\tau (x_1,x_2)\zeta ^{\prime \prime }_{x_2x_2}(x_1,x_2))\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)-\tau (x_1,x_2) \\ & \quad \times \bigg(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)-\zeta ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)\\ & \quad -\zeta ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)\\ & \quad -\tfrac {1}{2}\zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2) \\ & \quad -\zeta ^{\prime \prime }_{x_1x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2) \\ & \quad -\tfrac {1}{2}\zeta ^{\prime \prime }_{x_2x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)\bigg)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2) \\ & \quad \times \bigg(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\tau ;x_1,x_2) -\tau (x_1,x_2)-\tau ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2) \\ & \quad -\tau ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)\\ & \quad -\tfrac {1}{2}\tau ^{\prime \prime }_{x_1x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)\\ & \quad -\tau ^{\prime \prime }_{x_1x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2) \\ & \quad -\tfrac {1}{2}\tau ^{\prime \prime }_{x_2x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)\bigg)\\ & \quad +\tau ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2)(\zeta (x_1,x_2)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2))\\ & \quad +\tau ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2)(\zeta (x_1,x_2))-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2))\\ & \quad +\tau ^{\prime \prime }_{x_1x_1}(x_1,x_2)\tfrac {\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)}{2}(\zeta (x_1,x_2)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2))\\ & \quad +\tau ^{\prime \prime }_{x_2x_2}(x_1,x_2)\tfrac {\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)}{2}(\zeta (x_1,x_2)-\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2))\\ & \quad +\tau ^{\prime \prime }_{x_1x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1\! \! -\! \! x_1)(\hslash _2\! \! -\! \! x_2);x_1,x_2)\\ & \times (\zeta (x_1,x_2)\! \! -\quad -\! \! \mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2))\\ & \quad +\zeta ^{\prime }_{x_1}(x_1,x_2)\tau ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)\\ & \quad +\zeta ^{\prime }_{x_1}(x_1,x_2)\tau ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2)\\ & \quad +\zeta ^{\prime }_{x_2}(x_1,x_2)\tau ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2)\\ & \quad +\zeta ^{\prime }_{x_2}(x_1,x_2)\tau ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)\bigg). \end{align*}
</div>
<p>From <a href="#dc4t1">theorem 5.1</a>, for all \(\zeta \in \mathcal{C}(\mathcal{I}^2)\), it follows that \(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)\rightarrow \zeta (x_1,x_2)\), as \(n\rightarrow \infty \), uniformly in \((x_1,x_2)\in \mathcal{I}^2\), and by <a href="#dc4vo1">theorem 5.8</a>, for every \(\zeta \in \mathcal{C}^2(\mathcal{I}^2)\), we have </p>
<div class="displaymath" id="a0000000131">
  \begin{align*} & \lim _{n\rightarrow \infty }[n]_{q_n}\bigg[\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)- \\ & - \zeta ^{\prime }_{x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1);x_1,x_2) \\ & -\zeta ^{\prime }_{x_2}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2);x_1,x_2) \\ & -\tfrac {1}{2}\{ \zeta ^{\prime \prime }_{x_1x_1}(x_1,x_2)\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)\\ & \quad +2\zeta ^{\prime \prime }_{x_1x_2}\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2);x_1,x_2) \\ & \quad +\zeta ^{\prime \prime }_{x_2x_2}\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)\} \bigg]=0, \end{align*}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2.\) </p>
<p>Hence, in view of the fact that \(\zeta ,\tau \in \mathcal{C}^2(\mathcal{I}^2)\), using <a href="#dc4l4">lemma 4.3</a>, we reach the desired assertion. <div class="proof_wrapper" id="a0000000132">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <br /></p>
<h1 id="a0000000133">6 Construction of GBS operators for the bivariate operators \(\mathcal{G}^{\lowercase {q_{n_1}},\lowercase {q_{n_2}}}_{\lowercase {n_1},\lowercase {n_2},\alpha _1,\alpha _2}(\  \, \cdot \; ;\lowercase {x_1},\lowercase {x_2})\)</h1>
<p>In the past decade, the study of <i class="it">GBS</i> (Generalized Boolean Sum) operators associated with the positive linear operators has been an active area of research in the field of approximation theory. The concepts of Bögel continuous and Bögel differentiable functions were first given by Bögel <span class="cite">
	[
	<a href="#dc4Bogel1" >10</a>
	]
</span>. Dobrescu and Matei <span class="cite">
	[
	<a href="#DM" >17</a>
	]
</span> showed that any Bögel continuous function on a bounded interval can be uniformly approximated by the boolean sum of bivariate Bernstein polynomials. Badea <i class="it">et al.</i> <span class="cite">
	[
	<a href="#dc4Badea3" >6</a>
	]
</span> obtained a Korovkin type theorem for the Bögel continuous functions. Agrawal <i class="it">et al.</i> <span class="cite">
	[
	<a href="#dc4AP" >3</a>
	]
</span> studied convergence estimates for the GBS case of the bivariate Lupas-Durrmeyer type operators based on Polya distribution. Barbosu <i class="it">et al.</i> <span class="cite">
	[
	<a href="#BAM" >8</a>
	]
</span> proposed GBS operators of Durrmeyer-Stancu type based on \(q\)-integers and examined the approximation degree by using Lipschitz class and the mixed modulus of smoothness. Kajla and Miclaus <span class="cite">
	[
	<a href="#dc4AKD" >25</a>
	]
</span> determined the convergence behaviour of GBS operators of Bernstein-Durrmeyer type for Bögel continuous and Bögel differentiable functions. Agrawal and Chauhan <span class="cite">
	[
	<a href="#dc4RBP" >36</a>
	]
</span> introduced the sequence of GBS operators of Bernstein-Durrmeyer type on a triangle and investigated the rate of convergence by virtue of the mixed modulus of smoothness for Bögel continuous and Bögel differentiable functions. For a detailed account of the research in this direction, one can see <span class="cite">
	[
	<a href="#dc4ATPA" >19</a>
	]
</span> and the references therein. </p>
<p>A function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is called B-continuous (Bögel continuous) at a point \((x_1,x_2)\in \mathcal{I}^2\) if </p>
<div class="displaymath" id="a0000000134">
  \[ \lim _{(\hslash _1,\hslash _2)\rightarrow (x_1,x_2)}\Delta _{(x_1,x_2)}\zeta [(\hslash _1,\hslash _2);(x_1,x_2)]=0, \]
</div>
<p> where \(\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)]=\zeta (\hslash _1,\hslash _2)-\zeta (\hslash _1,x_2)-\zeta (x_1,\hslash _2)+\zeta (x_1,x_2).\) Further, a function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is said to be \(B\)- continuous on \(\mathcal{I}^2\), if is \(B\)- continuous \(\forall (x_1,x_2)\in \mathcal{I}^2\). </p>
<p>A function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is called \(B\)- differentiable (Bögel differentiable) on \(\mathcal{I}^2\), if for every \((x_1,x_2)\in \mathcal{I}^2\), </p>
<div class="displaymath" id="a0000000135">
  \[ \lim _{(\hslash _1,\hslash _2)\rightarrow (x_1,x_2)}\tfrac {\Delta _{(x_1,x_2)} \zeta [(\hslash _1, \hslash _2); (x_1,x_2)]}{(\hslash _1-x_1)(\hslash _2-x_2)}=D_{B}\zeta (x_1,x_2){\lt}\infty . \]
</div>
<p>The function \(\zeta :\mathcal{I}^2\rightarrow \mathbb {R}\) is said to be B-bounded on \(\mathcal{I}^2\) if \(\exists \) some \(K{\gt}0\), such that \(|\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2); (x_1, x_2)]|\leq K\), for every \((\hslash _1,\hslash _2), \, (x_1,x_2)\in \mathcal{I}^2\). The space of B-bounded functions is denoted by \(B_b( \mathcal{I}^2)\), the space of B-continuous functions is denoted by \(C_b( \mathcal{I}^2)\) and the space of all B-differentiable functions is denoted by \(D_b(\mathcal{I}^2).\) Further, let \(B(\mathcal{I}^2)\) be the space of bounded functions on \(\mathcal{I}^2\) endowed with the sup-norm denoted by \(\| .\| _{\infty }\). </p>
<p>For every \(\zeta \in \mathcal{C}_b(\mathcal{I}^2),\) the GBS operator associated with the operators defined in (<a href="#dc4abv1">4.2</a>) is defined as: </p>
<div class="displaymath" id="dc4abv2">
  \begin{align} \label{dc4abv2} \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)& =\sum _{k=0}^{n_1}\sum _{j=0}^{n_2} \mathfrak {P}^{\alpha _1,\alpha _2}_{n_1,n_2,k,j}(x_1,x_2) \bigg[\zeta \left(\tfrac {[k]_{q_{n_1}}}{[n_1]_{q_{n_1}}},x_2\right) \\ & \quad +\zeta \left(x_1,\tfrac {[j]_{q_{n_2}}}{[n_2]_{q_{n_2}}}\right)-\zeta \left(\tfrac {[k]_{q_{n_1}}}{[n_1]_{q_{n_1}}},\tfrac {[j]_{q_{n_2}}}{[n_2]_{q_{n_2}}}\right)\bigg].\nonumber \end{align}
</div>
<p>The operator \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) is well defined on the space \(\mathcal{C}_b(\mathcal{I}^2)\) into \(\mathcal{C}(\mathcal{I}^2)\). We shall analyze the order of approximation of \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \), for all \(\zeta \in \mathcal{C}_{b}(\mathcal{I}^2)\), using mixed modulus of smoothness. </p>
<p>First, we show the uniform convergence of the operators (<a href="#dc4abv2">6.11</a>) to \(\zeta \), where \(\zeta \in \mathcal{C}_{b}(\mathcal{I}^2)\) by using the following result: </p>
<p><div class="lemma_thmwrapper " id="dc4lbv67">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6.1</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#dc4BA" >7</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(\mathcal{K}_{m,n}:C_b(X\times Y)\rightarrow B(X\times Y), m, n\in \mathbb {N}\) be a sequence of bivariate positive linear operators. Further, let \(G_{m,n}\) be the associated GBS operators and the following identities hold: </p>
<ol class="enumerate">
  <li><p>\(\mathcal{K}_{m,n}(e_{00};x_1,x_2)=1;\) </p>
</li>
  <li><p>\(\mathcal{K}_{m,n}(e_{10};x_1,x_2)=x_1+\alpha _{m,n}(x_1,x_2);\) </p>
</li>
  <li><p>\(\mathcal{K}_{m,n}(e_{01};x_1,x_2)=x_2+\beta _{m,n}(x_1,x_2)\) </p>
</li>
  <li><p>\(\mathcal{K}_{m,n}(e_{20}+e_{02};x_1,x_2)=x_1^2+x_2^2+\gamma _{m,n}(x_1,x_2)\) </p>
</li>
</ol>
<p> for all \((x_1,x_2)\in X\times Y\). If the sequences \(\alpha _{m,n},\beta _{m,n}\), and \(\gamma _{m,n}\) converge to zero, as \(m,n\to \infty \), uniformly on \(X\times Y\), then the sequence \((G_{m,n}(\zeta ))\) converges to \(\zeta \), as \(m,n\to \infty \), uniformly on \(X\times Y\) for all \(\zeta \in C_b(X\times Y)\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="dc4lbv6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For \(\zeta \in \mathcal{C}_b(\mathcal{I}^2)\), we have </p>
<div class="displaymath" id="a0000000136">
  \begin{equation*}  \lim _{n\rightarrow \infty }\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)=\zeta (x_1,x_2), \end{equation*}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000137">
  <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="#dc4lbv2">lemma 4.1</a>, it follows that </p>
<div class="displaymath" id="a0000000138">
  \begin{equation*}  \lim _{n\rightarrow \infty }\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{ij})-e_{ij}\| _{\mathcal{C}(\mathcal{I}^2)}=0,\,  \forall \, (i,j)\, \in \, \{ (0,0),(1,0),(0,1)\}  \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000139">
  \begin{equation*}  \lim _{n\rightarrow \infty }\| \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{20}+e_{02})-(e_{20}+e_{02})\| _{\mathcal{C}(\mathcal{I}^2)}=0, \end{equation*}
</div>
<p> hence applying <a href="#dc4lbv67">lemma 6.1</a>, we obtain the desired conclusion. <div class="proof_wrapper" id="a0000000140">
  <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>In the next result, we examine convergence estimates of \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \) by virtue of mixed modulus of smoothness. </p>
<p>For \((x_1,x_2)\), \((\hslash _1,\hslash _2)\in \mathcal{I}^2\), the mixed modulus of smoothness of \(\zeta \in \mathcal{C}_b(\mathcal{I}^2)\) is defined by </p>
<div class="displaymath" id="dc4dg2">
  \begin{eqnarray} \label{dc4dg2} \widetilde{\omega _B}(\zeta ,\rho _1,\rho _2)=\sup \{ |\Delta _{(x_1,x_2)}\zeta [\hslash _1,\hslash _2;x_1,x_2]|:|\hslash _1-x_1|{\lt}\rho _1,|\hslash _2-x_2|{\lt}\rho _2\} \nonumber \end{eqnarray}
</div>
<p> for any \((\rho _1,\rho _2)\in (0,\infty )\times (0,\infty )\). From definition of \(\widetilde{\omega _B}(\zeta ,\rho _1,\rho _2)\), it follows that </p>
<div class="displaymath" id="a0000000141">
  \begin{equation*}  \widetilde{\omega _{B}}(\zeta ,c_1 \rho _1,c_2 \rho _2)\leq (1+c_1)(1+c_2)\widetilde{\omega _{B}}(\zeta ,\rho _1,\rho _2), \end{equation*}
</div>
<p> for any \(c_1,c_2{\gt}0\). </p>
<p><div class="theorem_thmwrapper " id="a0000000142">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For every \(\zeta \in \mathcal{C}_b(\mathcal{I}^2)\), and sufficiently large \(n_1\) and \(n_2\), the operator defined by <a href="#dc4abv2" class="eqref">6.11</a>, verifies the following result: </p>
<div class="displaymath" id="a0000000143">
  \begin{equation*}  \| \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )-\zeta \| _{\mathcal{C}(\mathcal{I}^2)}\leq K^{\alpha _2}_{\alpha _1}\widetilde{\omega _B}(\zeta ;[n_1]_{q_{n_1}}^{-\frac{1}{2}},[n_2]_{q_{n_2}}^{-\frac{1}{2}}),\nonumber \end{equation*}
</div>
<p> where \(K^{\alpha _2}_{\alpha _1}\) is some positive constant depending on \(\alpha _1\) and \(\alpha _2\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000144">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Considering the properties of the function \(\widetilde{\omega _B}\), we get </p>
<div class="displaymath" id="dc4dg3">
  \begin{align} \label{dc4dg3} |\Delta _{(x_1,x_2)}\zeta [(\hslash _1,\hslash _2);(x_1,x_2)]|& \leq \widetilde{\omega _B}(\zeta ;|\hslash _1-x_1|,|\hslash _2-x_2|)\nonumber \\ & \leq \bigg(1+\tfrac {|\hslash _1-x_1|}{\rho _1}\bigg)\bigg(1+\tfrac {|\hslash _2-x_2|}{\rho _2}\bigg)\, \widetilde{\omega _B}(\zeta ;\rho _1,\rho _2), \end{align}
</div>
<p> for every \((x_1,x_2),(\hslash _1,\hslash _2)\in \mathcal{I}^2\) and for any \(\rho _1, \rho _2 {\gt}0\). Using the definition of \(\Delta _{(x_1,x_2)}\zeta [(\hslash _1,\hslash _2);(x_1,x_2)]\), we may write </p>
<div class="displaymath" id="a0000000145">
  \begin{equation*}  \zeta (x_1,\hslash _2)+\zeta (\hslash _1,x_2)-\zeta (\hslash _1,\hslash _2)=\zeta (x_1,x_2)-\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)]. \end{equation*}
</div>
<p> Applying the operator \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of the above equality, we get </p>
<div class="displaymath" id="dc4dg4">
  \begin{align} \label{dc4dg4} \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)=& \zeta (x_1,x_2)\, \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(e_{00};x_1,x_2)\nonumber \\ & -\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\, (\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)];x_1,x_2).\quad \quad \end{align}
</div>
<p>Hence using <a href="#dc4dg3" class="eqref">6.11</a>, <a href="#dc4lbv2">lemma 4.1</a> and applying the Cauchy-Schwarz inequality, we get </p>
<div class="displaymath" id="dc4e10">
  \begin{align} \label{dc4e10} & |\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)|\leq \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\, (|\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)]|;x_1,x_2)\nonumber \\ & \leq \bigg(1+\rho _1^{-1}\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2;x_1,x_2)}\nonumber \\ & \quad +\rho _2^{-1}\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _2-x_2)^2;x_1,x_2)}\nonumber \\ & \quad +{\rho _1^{-1}}{\rho _2^{-1}}\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)}\bigg)\, \, \widetilde{\omega _B}(\zeta ;\rho _1,\rho _2).\nonumber \end{align}
</div>
<p> Now, choosing \(\rho _i=[n_i]_{q_{n_i}}^{\frac{-1}{2}}, i=1,2\), and applying <a href="#dc4l4">lemma 4.3</a>, the required assertion is proved. <div class="proof_wrapper" id="a0000000146">
  <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>The following result is concerned with the error in the approximation of the B-differentiable functions by the operators <a href="#dc4abv2" class="eqref">6.11</a>. </p>
<p><div class="theorem_thmwrapper " id="a0000000147">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(\zeta \in D_b(\mathcal{I}^2)\) and \(D_B\zeta \in B(\mathcal{I}^2),\) then </p>
<div class="displaymath" id="a0000000148">
  \begin{align*}  \| \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)& -\zeta (x_1,x_2)\| _{\mathcal{C}(\mathcal{I}^2)}\leq \\ & \leq \tfrac {C}{[n_1]^{\frac{1}{2}}_{q_{n_1}}[n_2]^{\frac{1}{2}}_{q_{n_2}}} \bigg\{ \| D_B \zeta \| _{\infty }+ \widetilde{\omega _B}(D_{b}\zeta ;[n_1]_{q_{n_1}}^{-\frac{1}{2}},{[n_2]_{q_{n_2}}^{-\frac{1}{2}}})\bigg\} , \end{align*}
</div>
<p> where \(C{\gt}0\) is some constant. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000149">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(\zeta \in D_b(\mathcal{I}^2)\), by mean value theorem, we have </p>
<div class="displaymath" id="a0000000150">
  \begin{equation*}  \Delta _{(x_1,x_2)}\zeta [(\hslash _1,\hslash _2);(x_1,x_2)]=(\hslash _1-x_1)(\hslash _2-x_2) D_B \zeta (\alpha ,\beta ),\,  with\,  x_1{\lt}\alpha {\lt}\hslash _1, \end{equation*}
</div>
<p> and \(\, x_2{\lt}\beta {\lt}\hslash _2.\) Clearly, </p>
<div class="displaymath" id="a0000000151">
  \begin{equation*}  D_B \zeta (\alpha ,\beta )\! \! =\! \! \Delta _{(x_1,x_2)} D_B \zeta [(\alpha ,\beta );(x_1,x_2)]\! \! +\! \! D_B \zeta (\alpha ,x_2)\! +\! D_B \zeta (x_1,\beta )\! -\! D_B \zeta (x_1,x_2). \end{equation*}
</div>
<p> Since \(D_B \zeta \in B(\mathcal{I}^2)\), from the above equalities, we have </p>
<div class="displaymath" id="dc4e11">
  \begin{align} \label{dc4e11} & |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)];x_1,x_2)|= \\ & = |\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)(\hslash _2-x_2) D_B \zeta (\alpha ,\beta );x_1,x_2)]|\nonumber \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1||\hslash _2-x_2||\Delta _{(x_1,x_2)} D_B \zeta [(\alpha ,\beta ); (x_1,x_2)]|;x_1,x_2)\nonumber \\ &  \quad +\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\Big(|\hslash _1-x_1||\hslash _2-x_2|\Big(|D_B \zeta (\alpha ,x_2)| \nonumber \\ & \hspace{5cm}+|D_B \zeta (x_1,\beta )|+|D_B \zeta (x_1,x_2)|\Big);x_1,x_2\Big)\nonumber \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1||\hslash _2-x_2|\widetilde{\omega _B}(D_B\zeta ;|\alpha -x_1|,|\beta -x_2|);x_1,x_2)\nonumber \\ & \quad +3\| |D_B \zeta ||_{\infty }\, \,  \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1||\hslash _2-x_2|;x_1,x_2).\nonumber \end{align}
</div>
<p> Considering the properties of mixed modulus of smoothness \(\widetilde{\omega _B}\), for any \(\rho _1,\\ \rho _2 {\gt}0,\) we have </p>
<div class="displaymath" id="dc4e12">
  \begin{align} \label{dc4e12} \widetilde{\omega _B}(D_B \zeta ;|\alpha -x_1|,|\beta -x_2|)& \leq \widetilde{\omega _B}(D_B \zeta ;|\hslash _1-x_1|,|\hslash _2-x_2|)\nonumber \\ & \leq \prod _{i=1}^2 (1+\rho ^{-1}_i|\hslash _i-x_i|)\, \, \widetilde{\omega _B}(D_B \zeta ;\rho _1,\rho _2).\end{align}
</div>
<p> Hence taking into account (<a href="#dc4e11">6.14</a>), (<a href="#dc4e12">6.15</a>) and applying the Cauchy-Schwarz inequality, we obtain </p>
<div class="displaymath" id="dc4a1">
  \begin{align} \label{dc4a1} & |\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)-\zeta (x_1,x_2)|= \\ & =|\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\Delta \zeta [(\hslash _1,\hslash _2);(x_1,x_2)];x_1,x_2)|\leq \nonumber \\ & \leq 3||D_B \zeta ||_\infty \sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)}\nonumber \\ & \quad +\bigg(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1||\hslash _2-x_2|;x_1,x_2)\nonumber \\ & \quad +\rho ^{-1}_1 \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2|\hslash _2-x_2|;x_1,x_2)\nonumber \\ & \quad +\rho ^{-1}_2\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(|\hslash _1-x_1|(\hslash _2-x_2)^2;x_1,x_2)\nonumber \\ & \quad +{\rho ^{-1}_1}{\rho ^{-1}_2}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)\bigg)\widetilde{\omega _B}(D_B \zeta ;\rho _1,\rho _2)\nonumber \\ & \leq 3||D_B \zeta ||_\infty \sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)}\nonumber \\ & \quad +\bigg(\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)}\nonumber \\ & \quad +\rho ^{-1}_1\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^4(\hslash _2-x_2)^2;x_1,x_2)}\nonumber \\ & \quad +\rho ^{-1}_2\sqrt{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^4;x_1,x_2)}\nonumber \\ & \quad +{\rho ^{-1}_1}{\rho ^{-1}_2}\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^2(\hslash _2-x_2)^2;x_1,x_2)\bigg)\widetilde{\omega _B}(D_B \zeta ;\rho _1,\rho _2).\nonumber \end{align}
</div>
<p> Since, for \(i,j\in \{ 1,2\} ,\) we have </p>
<div class="displaymath" id="dc4a2">
  \begin{align} \label{dc4a2} & \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}((\hslash _1-x_1)^{2i}(\hslash _2-x_2)^{2j};x_1,x_2)=\nonumber \\ & ={\mathcal O}(\tfrac {1}{[n_1]_{q_{n_1}}^{i}}){\mathcal O}(\tfrac {1}{[n_2]_{q_{n_2}}^{j}}), \qquad as \, n_1,n_2\rightarrow \infty , \end{align}
</div>
<p> uniformly in \((x_1,x_2)\in \mathcal{I}^2\), combining <a href="#dc4a1" class="eqref">6.16</a>–<a href="#dc4a2" class="eqref">6.17</a> and choosing \(\rho _i=[n_i]_{q_{n_i}}^{\frac{-1}{2}},i=1,2\), we reach the desired result. <div class="proof_wrapper" id="a0000000152">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, we discuss the approximation degree of the operators <a href="#dc4abv2" class="eqref">6.11</a> for Lipschitz class of B-continuous functions. </p>
<p>The Lipschitz class \(\operatorname {Lip}_{\mathcal{M},b}^{\theta }\) with \(\theta \in (0,1]\), for B-continuous functions is defined by </p>
<div class="displaymath" id="a0000000153">
  \begin{equation*}  \operatorname {Lip}_{\mathcal{M},b}^{\theta } \! \! =\! \! \left\{  \zeta \in \mathcal{C}_b(\mathcal{I}^2)\! :\! |\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)]|\! \leq \!  \mathcal{M}\{ (\hslash _1\! \! -\! \! x_1)^2\! +\! (\hslash _2\! \! -\! \! x_2)^2\} ^{\frac{\theta }{2}}\right\}  \end{equation*}
</div>
<p> for every \(\, (\hslash _1,\hslash _2),\, (x_1,x_2) \in \mathcal{I}^2.\) <div class="theorem_thmwrapper " id="a0000000154">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For \(\zeta \in \operatorname {Lip}_{\mathcal{M},b}^{\theta }\), \(\theta \in (0,1]\), we have </p>
<div class="displaymath" id="a0000000155">
  \begin{equation*}  \| \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ) -\zeta \| _{\mathcal{C}(\mathcal{I}^2)} \leq \mathcal{M}(\theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2})^{\frac{\theta }{2}}. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000156">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Considering <a href="#dc4lbv2">lemma 4.1</a> and <a href="#dc4dg4" class="eqref">6.12</a>, by our hypothesis we get </p>
<div class="displaymath" id="a0000000157">
  \begin{align*} & \left\vert \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}( \zeta ;x_1,x_2) -\zeta (x_1,x_2)\right\vert \leq \\ & \leq \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left( \left\vert \Delta _{(\hslash _1,\hslash _2)} \zeta [(x_1,x_2);(x_1,x_2)]\right\vert ;x_1,x_2\right) \\ & \leq \mathcal{M} \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left(( (\hslash _1-x_1)^2+(\hslash _2-x_2)^2)^{\frac{\theta }{2}};x_1,x_2\right). \end{align*}
</div>
<p>Now, applying the Hölder’s inequality with \((p_{1},q_{1})=\left(\tfrac {2}{\theta } ,\tfrac {2}{( 2-\theta )}\right) \) and using <a href="#dc4lbv2">lemma 4.1</a>, we get </p>
<div class="displaymath" id="a0000000158">
  \begin{align*} & | \mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}( \zeta ;x_1,x_2) -\zeta (x_1,x_2)|\leq \\ & \leq \{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\left( e_{00};x_1,x_2\right)\} ^{\frac{2}{( 2-\theta )}} \\ & \quad \times \mathcal{M} \{ \mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2} \left(((\hslash _1-x_1)^{2}+(\hslash _2-x_2)^2);x_1,x_2\right)\} ^{\theta /2}\\ & \leq \mathcal{M}(\theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2})^{\frac{\theta }{2}}, \end{align*}
</div>
<p> which yields us the required result. This completes the proof. <div class="proof_wrapper" id="a0000000159">
  <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="acknowledgement_thmwrapper " id="a0000000160">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The second author is thankful to "The Ministry of Human Resource and Development", India for the financial support to carry out the above work. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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  <dd><p><i class="sc">R.A. Devore</i> and <i class="sc">G.G. Lorentz</i>, <i class="it">Constructive Approximation</i>, <b class="bf">303</b>, Springer Science and Business Media, 1993. </p>
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  <dd><p><a href ="https://doi.org/10.1007/BF01146928"> <i class="sc">A.D. Gadjiev</i>, <i class="it">Theorems of Korovkin type</i>, Mat. Zametki, <b class="bf">20</b> (1976) no. 5, 781–786 (in Russian), Math. Notes, <b class="bf">20</b> (1976) nos. 5–6, 995–998 (Engl. Trans.)). <a href="https://doi.org/10.1007/BF01146928">https://doi.org/10.1007/BF01146928</a> <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><p><i class="sc">V. Gupta</i> and <i class="sc">R.P. Agarwal</i>, <i class="it">Convergence Esimates in Approximation Theory</i>, Springer, Berlin (2014). </p>
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  <dd><p><a href ="https://doi.org/10.1007/s11075-018-0538-7"> <i class="sc">V. Gupta</i>, <i class="sc">G. Tachev</i>, <i class="sc">A.M. Acu</i>, <i class="it">Modified Kantorovich operators with better approximation properties</i>, Numer. Algor., <b class="bf">81</b> (2019) no. 1, 125–149. <a href="https://doi.org/10.1007/s11075-018-0538-7">https://doi.org/10.1007/s11075-018-0538-7</a> <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><p><a href ="https://doi.org/10.2478/s11533-009-0055-y"> <i class="sc">V. Gupta</i>, <i class="sc">C. Radu</i>, <i class="it">Statistical approximation properties of q-Baskakov-Kantorovich operators</i>, Cent. Eur. J. Math., <b class="bf">7</b> (2009) no. 4, 809–818. 809-818 <a href="https://doi.org/10.2478/s11533-009-0055-y">https://doi.org/10.2478/s11533-009-0055-y</a> <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><p><a href ="https://doi.org/10.1007/s00025-018-0773-1"> <i class="sc">A. Kajla</i> and <i class="sc">D. Miclaus</i>, <i class="it">Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type</i>, Results Math., <b class="bf">73</b> (2018) no. 1, 1–21. <a href="https://doi.org/10.1007/s00025-018-0773-1">https://doi.org/10.1007/s00025-018-0773-1</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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</a> </p>
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  <dt><a name="NIM">30</a></dt>
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</a> </p>
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  <dt><a name="MMA">32</a></dt>
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</a> </p>
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</a> </p>
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  <dd><p><a href ="https://doi.org/10.1016/S0377-0427(02)00733-1"> <i class="sc">H. Oruc</i>, <i class="sc">G.M. Phillips</i>, <i class="it">q-Bernstein polynomials and Bézier curves</i>, J. Comput. Appl. Math., <b class="bf">151</b> (2003), 1–12. <a href="https://doi.org/10.1016/S0377-0427(02)00733-1">https://doi.org/10.1016/S0377-0427(02)00733-1</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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</a> </p>
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  <dd><p><a href ="https://doi.org/10.1002/mma.4771"> <i class="sc">R. Ruchi</i>, <i class="sc">B. Baxhaku</i>, <i class="sc">P.N. Agrawal</i>, <i class="it">GBS operators of bivariate Bernstein-Durrmeyer-type on a triangle</i>, Math. Methods Appl. Sci., <b class="bf">41</b> (2018) no. 7, 2673–2683. <a href="https://doi.org/10.1002/mma.4771">https://doi.org/10.1002/mma.4771</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dt><a name="dc4VIK">37</a></dt>
  <dd><p><i class="sc">V.I. Volkov</i>, <i class="it">On the convergence of sequences of linear positive operators in the space of continuous functions of two variables</i>, Dolk. Akad. Nauk, SSRR, <b class="bf">115</b> (1957) no. 1, 17–19. </p>
</dd>
</dl>


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