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<title>The generalization of some results for Schurer and Schurer-Stancu operators: The generalization of some results for Schurer and Schurer-Stancu operators</title>
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<h1>The generalization of some results for Schurer and Schurer-Stancu operators</h1>
<p class="authors">
<span class="author">Dan Miclăuş\(^\ast \)</span>
</p>
<p class="date">January 27, 2011</p>
</div>
<p>\(^\ast \) Department of Mathematics and Computer Science, Faculty of Sciences, North University of Baia Mare, Victoriei 76, 430122 Baia Mare, Romania, e-mail: <span class="tt">danmiclausrz@yahoo.com</span> </p>

<div class="abstract"><p> In the present paper we generalize some results for Schurer and Schurer-Stancu operators. Firstly, we establish a general formula concerning calculation of test functions by Schurer operators. Secondly, using this relationship and some known results we prove in every case a Voronovskaja type theorem, the uniform convergence and the order of approximation for Schurer and Schurer-Stancu operators. </p>
<p><b class="bf">MSC.</b> 41A10, 41A25, 41A36 </p>
<p><b class="bf">Keywords.</b> Bernstein operators, Schurer operators, Schurer-Stancu operators, falling factorials, Pochhammer symbol, Stirling numbers of second kind, Voronovskaja type theorem, modulus of continuity, order of approximation </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\mathbb {N}\) be the set of positive integers and \(\mathbb {N}_{0}=\mathbb {N}\cup \{ 0\} .\) The operators \(B_{n}:C([0,1])\to C([0,1])\) given by </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} B_{n}(f;x)=\sum \limits _{k=0}^{n}p_{n,k}(x)f\left(\tfrac {k}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p> where \(p_{n,k}(x)\) are the fundamental Bernstein’s polynomials defined by </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.2} p_{n,k}(x)=\tbinom {n}{k}x^{k}(1-x)^{n-k}, \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p> for any \(x\in [0,1]\), \(k\in \{ 0,1,\ldots ,n\} \) and \(n\in \mathbb {N}\), are called Bernstein operators and were first introduced in <span class="cite">
	[
	<a href="#SNB" >8</a>
	]
</span>. In what follows, let \(p\in \mathbb {N}_{0}\) be a fixed natural number and let the real parameters \(\alpha , \beta \) be given such that \(0\leq \alpha \leq \beta \). The operators \(\tilde{B}_{n,p}:C([0,1+p])\to C([0,1])\) given by </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.3} \tilde{B}_{n,p}(f;x)=\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)f\left(\tfrac {k}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> where \(\tilde{p}_{n,k}(x)\) are the fundamental Schurer’s polynomials defined by </p>
<div class="equation" id="1.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.4} \tilde{p}_{n,k}(x)=\tbinom {n+p}{k}x^k(1-x)^{n+p-k}, \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p> for any \(x\in [0,1]\), \(k\in \{ 0,1,\ldots ,n+p\} \) and \(n\in \mathbb {N}\), are called Schurer operators <span class="cite">
	[
	<a href="#FS" >20</a>
	]
</span>. The operators \(\tilde{S}_{n,p}^{(\alpha ,\beta )}:C([0,1+p])\to C([0,1])\) defined by </p>
<div class="equation" id="1.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.5} \tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)=\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)f\left(\tfrac {k+\alpha }{n+\beta }\right), \end{equation}
  </div>
  <span class="equation_label">1.5</span>
</p>
</div>
<p> for any \(x\in [0,1]\), \(k\in \{ 0,1,\ldots ,n+p\} \) and \(n\in \mathbb {N}\), where \(\tilde{p}_{n,k}(x)\) are the fundamental Schurer’s polynomials given at <a href="#1.4" class="eqref">1.4</a>, are called Schurer-Stancu operators and were first introduced in <span class="cite">
	[
	<a href="#HHGJM" >9</a>
	]
</span>, then studied intensively in <span class="cite">
	[
	<a href="#DB2" >6</a>
	]
</span>. </p>
<p><div class="remark_thmwrapper " id="a0000000003">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.1</span>
  </div>
  <div class="remark_thmcontent">
  <p>More results and properties concerning <a href="#1.3" class="eqref">1.3</a> and <a href="#1.5" class="eqref">1.5</a> can be found also in monographs <span class="cite">
	[
	<a href="#OA" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#FAMC" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#DB3" >7</a>
	]
</span>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>The aim of this paper is to generalize some results for the presented operators. Firstly, we establish a general formula concerning calculation of the test functions by Schurer operators and next, taking this into account, we will prove a Voronovskaja type theorem in every case for Schurer and Schurer-Stancu operators. Using some known results, which will be cited at the adequate moment we shall prove the uniform convergence, general Voronovskaja type formulas and the order of approximation up to twice continuously differentiable function for the Bernstein type operators. </p>
<h1 id="a0000000004">2 Preliminaries</h1>
<p>Of the greatest utility in the calculus of finite differences, in number theory, in the summation of series, in the calculation of the Bernstein polynomials are the numbers introduced in 1730 by J. Stirling in his <i class="itshape">Methodus differentialis</i> <span class="cite">
	[
	<a href="#JS" >21</a>
	]
</span>, subsequently called “Stirling numbers" of the first and second kind. For any \(x\in \mathbb {R}\) and \(n\in \mathbb {N}_0\), let </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  (x)_n:=\prod \limits _{i=0}^{n-1}(x-i), \end{equation*}
</div>
<p> where \((x)_0:=1\) is the falling factorial denoted by Pochhammer symbol. It is well known that </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} x^j=\sum \limits _{i=0}^jS(j,i)(x)_i \end{equation}
  </div>
  <span class="equation_label">2.6</span>
</p>
</div>
<p> holds for any \(x\in \mathbb {R}\) and \(j\in \mathbb {N}_0,\) where \(S(j,i)\) are the Stirling numbers of second kind. Now, let \(i,j\in \mathbb {N}_0\) be natural numbers, then the Stirling numbers of second kind have the following properties: </p>
<div class="equation" id="2.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.2} S(j,i):=\begin{cases} 1, & \text{if}\  j=i=0;\  j=i\  \text{or} j>1,i=1\\ 0, &  \text{if}\  j>0,i=0\\ 0, &  \text{if}\  j<i\\ i\cdot S(j\! -\! 1,i)\! +\! S(j\! -\! 1,i\! -\! 1), &  \text{if}\  j,i>1. \end{cases} \end{equation}
  </div>
  <span class="equation_label">2.7</span>
</p>
</div>
<p> Let \(e_j(x)=x^j\), with \(j\in \mathbb {N}_0\) be the test functions. The main result established in <span class="cite">
	[
	<a href="#OTPMF" >16</a>
	]
</span>, by O.T. Pop and M. Farcaş concerning calculation of the test functions in general case by Bernstein operators is given by the following: </p>
<p><div class="proposition_thmwrapper " id="a0000000006">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2.1</span>
  </div>
  <div class="proposition_thmcontent">
  <p><span class="cite">
	[
	<a href="#OTPMF" >16</a>
	]
</span> If \(n,j\in \mathbb {N}\), then </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} B_n(e_j;x)=\tfrac {1}{n^j}\sum \limits _{i=1}^jS(j,i)(n)_ix^i. \end{equation}
  </div>
  <span class="equation_label">2.12</span>
</p>
</div>

  </div>
</div> During the preparation of the present paper, making some researches we discovered that, the relation <a href="#2.3" class="eqref">2.12</a> had been proved earlier by S. Karlin and Z. Ziegler <span class="cite">
	[
	<a href="#SKZZ" >13</a>
	]
</span>. As a special case, we can find the same relation in the article <span class="cite">
	[
	<a href="#UAMI" >1</a>
	]
</span>, where the asymptotic expansion of multivariate Bernstein polynomials on a simplex are considered. Later, in <span class="cite">
	[
	<a href="#OTPDBPIB" >19</a>
	]
</span> the authors O.T. Pop, D. Bărbosu and P.I. Braica proved another result concerning calculation of the test functions by Bernstein operators. In <span class="cite">
	[
	<a href="#DMPIB" >14</a>
	]
</span> we established that the result proved in <span class="cite">
	[
	<a href="#OTPDBPIB" >19</a>
	]
</span> does not differ by the result given at <a href="#2.3" class="eqref">2.12</a>. </p>
<p>In this section we recall some results from <span class="cite">
	[
	<a href="#OTP1" >17</a>
	]
</span> and <span class="cite">
	[
	<a href="#OTP2" >18</a>
	]
</span>, which we shall use in the present paper. Let \(I\), \(J\) be real intervals and \(I\cap J\neq \emptyset \). For any \(n,k\in \mathbb {N}_{0}\), \(n\neq 0\) consider the functions \(\varphi _{n,k}:J\to \mathbb {R},\) with the property that \(\varphi _{n,k}(x)\geq 0\), for any \(x\in J\) and also consider the linear positive functionals \(A_{n,k}:E(I)\to \mathbb {R}.\) For any \(n\in \mathbb {N}\) define the operator \(L_{n}:E(I)\to F(J)\), by </p>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} L_{n}(f;x)=\sum \limits _{k=0}^{n}\varphi _{n,k}(x)A_{n,k}(f), \end{equation}
  </div>
  <span class="equation_label">2.13</span>
</p>
</div>
<p> where \(E(I)\) is a linear space of real-valued functions defined on \(I\) and \(F(J)\) is a subset of the set of real-valued functions defined on \(J\). </p>
<p><div class="remark_thmwrapper " id="a0000000007">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.2</span>
  </div>
  <div class="remark_thmcontent">
  <p><span class="cite">
	[
	<a href="#OTP1" >17</a>
	]
</span> The operators \((L_{n})_{n\in \mathbb {N}}\) are linear and positive on <br />\(E(I\cap J)\).<span class="qed">â–¡</span></p>

  </div>
</div> For \(n\in \mathbb {N}\) and \(i\in \mathbb {N}_{0}\) define \(T^{\ast }_{n,i}\) by </p>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.5} T^{\ast }_{n,i}(L_n;x)=n^iL_{n}\left(\psi _{x}^{i};x\right)=n^i\sum \limits _{k=0}^{n}\varphi _{n,k}(x)A_{n,k}\left(\psi _{x}^{i}\right),\  \  \  x\in I\cap J, \end{equation}
  </div>
  <span class="equation_label">2.14</span>
</p>
</div>
<p> where \(\psi _x^i=(t-x)^i\), \(t\in I\cap J\). </p>
<p>In what follows \(s\in \mathbb {N}_{0}\) is even and we assume that the next two conditions hold: </p>
<ul class="itemize">
  <li><p>there exists the smallest \(\alpha _{s},\  \alpha _{s+2}\in [0,+\infty )\), so that </p>
<div class="equation" id="2.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.6} \lim \limits _{n\to \infty }\frac{T^{\ast }_{n,j}(L_{n};x)}{n^{\alpha _{j}}}=B_{j}(x)\in \mathbb {R}, \end{equation}
  </div>
  <span class="equation_label">2.15</span>
</p>
</div>
<p> for any \(x\in I\cap J\) and \(j\in \{ s,s+2\} \), </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} \alpha _{s+2}<\alpha _{s}+2 \end{equation}
  </div>
  <span class="equation_label">2.16</span>
</p>
</div>
</li>
  <li><p>\(I\cap J\) is an interval. </p>
</li>
</ul>
<p><div class="theorem_thmwrapper " id="th1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> <span class="cite">
	[
	<a href="#OTP1" >17</a>
	, 
	<a href="#OTP2" >18</a>
	]
</span> If \(f\in E(I)\) is a function \(s\) times differentiable in a neighborhood of \(x\in I\cap J\), then </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} \lim \limits _{n\to \infty }n^{s-\alpha _{s}}\left(L_{n}(f;x)-\sum \limits _{i=0}^{s}\tfrac {f^{(i)}(x)}{i!\cdot n^{i}}T^{\ast }_{n,i}(L_{n};x)\right)=0. \end{equation}
  </div>
  <span class="equation_label">2.17</span>
</p>
</div>
<p> Assume that \(f\) is \(s\) times differentiable function on \(I\) and there exists an interval \(K\subseteq I\cap J\), such that, there exist \(n(s)\in \mathbb {N}\) and the constants \(k_{j}\in \mathbb {R}\) depending on \(K\), so that for \(n\geq n(s)\) and \(x\in K\), the following </p>
<div class="equation" id="2.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.9} \frac{T^{\ast }_{n,j}(L_{n};x)}{n^{\alpha _{j}}}\leq k_{j}, \end{equation}
  </div>
  <span class="equation_label">2.18</span>
</p>
</div>
<p> holds, for \(j\in \{ s,s+2\} \). </p>
<p>Then, the convergence expressed by <a href="#2.8" class="eqref">2.17</a> is uniform on \(K\) and moreover </p>
<div class="displaymath" id="2.10">
  \begin{align} \label{2.10} & n^{s-\alpha _{s}} \left|L_{n}(f;x)-\sum \limits _{i=0}^{s}\tfrac {f^{(i)}(x)}{i!\cdot n^{i}}T^{\ast }_{n,i}(L_{n};x)\right|\leq \\ & \leq \tfrac {1}{s!}(k_{s}+k_{s+2})\omega _{1}\left(f^{(s)};\tfrac {1}{\sqrt{n^{2+\alpha _{s}-\alpha _{s+2}}}}\right),\nonumber \end{align}
</div>
<p> for any \(x\in K\) and \(n\geq n(s)\), where \(\omega _{1}(f;\delta )\) denotes the modulus of continuity of the function \(f\). </p>

  </div>
</div> </p>
<h1 id="a0000000008">3 Main results</h1>
<p>In the case of Schurer operators, we get: <div class="proposition_thmwrapper " id="a0000000009">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3.1</span>
  </div>
  <div class="proposition_thmcontent">
  <p>For any \(j,n\in \mathbb {N}\) and \(x\in [0,1]\), the following holds </p>
<div class="equation" id="3.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.1} \tilde{B}_{n,p}(e_j;x)=\tfrac {1}{n^j}\sum \limits _{i=0}^{j-1}S(j,j-i)(n+p)_{j-i}x^{j-i}. \end{equation}
  </div>
  <span class="equation_label">3.20</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For the proof of this proposition we take into account the same idea used in <span class="cite">
	[
	<a href="#OTPMF" >16</a>
	]
</span>. The relation <a href="#2.1" class="eqref">2.6</a> can be written also in the following form </p>
<div class="equation" id="3.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.2} x^j=\sum \limits _{i=0}^{j-1}S(j,j-i)(x)_{j-i}, \end{equation}
  </div>
  <span class="equation_label">3.21</span>
</p>
</div>
<p> because \(S(j,0)=0\), see <a href="#2.2" class="eqref">2.7</a>. Using <a href="#3.2" class="eqref">3.21</a>, we get </p>
<div class="displaymath" id="a0000000011">
  \begin{align*}  \tilde{B}_{n,p}(e_j;x)& =\sum \limits _{k=0}^{n+p}\tbinom {n+p}{k}x^k(1-x)^{n+p-k}\left(\tfrac {k}{n}\right)^j\\ & =\tfrac {1}{n^j}\sum \limits _{k=0}^{n+p}\tbinom {n+p}{k}x^k(1-x)^{n+p-k}\sum \limits _{i=0}^{j-1}S(j,j-i)(k)_{j-i}\\ & =\tfrac {1}{n^j}\sum \limits _{i=0}^{j-1}S(j,j-i)\sum \limits _{k=0}^{n+p}(k)_{j-i}\tbinom {n+p}{k}x^k(1-x)^{n+p-k}= \end{align*}
</div>
<div class="displaymath" id="a0000000012">
  \[ =\tfrac {1}{n^j}\sum \limits _{i=0}^{j-1}S(j,j-i)(n+p)_{j-i}x^{j-i}. \]
</div>
<p> <div class="proof_wrapper" id="a0000000013">
  <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 following, we assume that the first three cases concerning calculation of the test functions by Schurer, respectively Schurer-Stancu operators are well known and for more details we recommend the reader our paper <span class="cite">
	[
	<a href="#DMPIB1" >15</a>
	]
</span>. </p>
<h2 id="a0000000014">3.1 Schurer operators</h2>
<p> Using the construction form preliminaries, we assume that \(I=[0,1+p]\), \(J=[0,1]\), \(E(I)=C([0,1+p])\), \(F(J)=C([0,1])\) and the role of \(n\) is played by \(n+p\). Then let the functions \(\varphi _{n+p,k}:[0,1]\to \mathbb {R}\) be defined by \(\varphi _{n+p,k}(x):=\tilde{p}_{n,k}(x)\), for any \(x\in [0,1]\), \(n,k\in \mathbb {N}_0\), \(n\neq 0\) and the functionals \(A_{n+p,k}:C([0,1+p])\to \mathbb {R}\) let be defined by \(A_{n+p,k}(f):=f\left(\tfrac {k}{n}\right)\), for any \(n,k\in \mathbb {N}_0\), \(n\neq 0\). In this case one obtains the Schurer operators, with </p>
<div class="displaymath" id="3.3">
  \begin{align} \label{3.3} T^{\ast }_{n,i}\left(\tilde{B}_{n,p};x\right)& =n^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)A_{n+p,k}\left(\psi _x^i\right)\\ & \nonumber =n^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)\left(\tfrac {k}{n}-x\right)^i\\ & \nonumber =\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)(k-(n+p)x+px)^i\\ & \nonumber =\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)\sum \limits _{l=0}^i\tbinom {i}{l}(k-(n+p)x)^l(px)^{i-l}\\ & \nonumber =\sum \limits _{l=0}^i\tbinom {i}{l}(px)^{i-l}T_{n+p,l}(x),\nonumber \end{align}
</div>
<p> where </p>
<div class="equation" id="3.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.4} T_{n+p,l}(x)=\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)(k-(n+p)x)^l. \end{equation}
  </div>
  <span class="equation_label">3.22</span>
</p>
</div>
<p><div class="application_thmwrapper " id="ap">
  <div class="application_thmheading">
    <span class="application_thmcaption">
    Application
    </span>
    <span class="application_thmlabel">3.2</span>
  </div>
  <div class="application_thmcontent">
  <p> For \(j\in \{ 3,4\} \) we present the calculation of the test functions by Schurer operators, taking into account <a href="#3.1" class="eqref">3.20</a>. </p>
<p>Case 1. \(j=3\) </p>
<div class="displaymath" id="a0000000015">
  \begin{align*}  \tilde{B}_{n,p}(e_3;x)& =\tfrac {1}{n^3}\sum \limits _{i=0}^2S(3,3-i)(n+p)_{3-i}x^{3-i}\\ & =\tfrac {1}{n^3}\left(S(3,3)(n+p)_3x^3+S(3,2)(n+p)_2x^2+S(3,1)(n+p)_1x\right)\\ & =\tfrac {1}{n^3}\left((n+p)_3x^3+3(n+p)_2x^2+(n+p)_1x\right), \end{align*}
</div>
<p> where \(S(3,2)=2\cdot S(2,2)+S(2,1)=3.\) </p>
<p>Case 2. \(j=4\) </p>
<div class="displaymath" id="a0000000016">
  \begin{align*}  \tilde{B}_{n,p}(e_4;x)& =\tfrac {1}{n^4}\sum \limits _{i=0}^3S(4,4-i)(n+p)_{4-i}x^{4-i}\\ & =\tfrac {1}{n^4}\left(S(4,4)(n+p)_4x^4\right.\\ & \quad \left.+S(4,3)(n+p)_3x^3+S(4,2)(n+p)_2x^2+S(4,1)(n+p)_1x\right)\\ & =\tfrac {1}{n^4}\left((n+p)_4x^4+6(n+p)_3x^3+7(n+p)_2x^2+(n+p)_1x\right), \end{align*}
</div>
<p> where \(S(4,2)=2\cdot S(3,2)+S(3,1)=7\) and \(S(4,3)=3\cdot S(3,3)+S(3,2)=6.\) </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000017">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>Regarding the polynomials \(T_{n+p,l}(x)\), which were first introduced in <span class="cite">
	[
	<a href="#DBMB" >4</a>
	]
</span>, we shall give a proof relied on Application <a href="#ap">3.2</a>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.4</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The polynomials \(T_{n+p,l}(x)\) satisfy the following </p>
<div class="displaymath" id="a0000000018">
  \begin{align*}  T_{n+p,0}(x)& =1,\\ T_{n+p,1}(x)& =0,\\ T_{n+p,2}(x)& =(n+p)x(1-x),\\ T_{n+p,3}(x)& =(n+p)x(1-x)(1-2x),\\ T_{n+p,4}(x)& =3(n+p)^2x^2(1-x)^2+(n+p)\left(x(1-x)-6x^2(1-x)^2\right). \end{align*}
</div>

  </div>
</div> <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> Using <a href="#3.4" class="eqref">3.22</a> and Application <a href="#ap">3.2</a>, it follows </p>
<div class="displaymath" id="a0000000020">
  \begin{align*}  T_{n+p,0}(x)& =\tilde{B}_{n,p}(e_0;x)=1;\\ T_{n+p,1}(x)& =n\tilde{B}_{n,p}(e_1;x)-(n+p)x\tilde{B}_{n,p}(e_0;x)=0;\\ T_{n+p,2}(x)& =n^2\tilde{B}_{n,p}(e_2;x)-2n(n+p)x\tilde{B}_{n,p}(e_1;x)+((n+p)x)^2\tilde{B}_{n,p}(e_0;x)\\ & =(n+p)x(1-x);\\ T_{n+p,3}(x)& =n^3\tilde{B}_{n,p}(e_3;x)-3n^2(n+p)x\tilde{B}_{n,p}(e_2;x)+3n((n+p)x)^2\tilde{B}_{n,p}(e_1;x)\\ & \quad -((n+p)x)^3\tilde{B}_{n,p}(e_0;x)=(n+p)x(1-x)(1-2x); \end{align*}
</div>
<div class="displaymath" id="a0000000021">
  \begin{align*}  T_{n+p,4}(x)& =n^4\tilde{B}_{n,p}(e_4;x)-4n^3(n+p)x\tilde{B}_{n,p}(e_3;x)+6(nx(n+p))^2\tilde{B}_{n,p}(e_2;x)\\ & \quad -4n((n+p)x)^3\tilde{B}_{n,p}(e_1;x)+((n+p)x)^4\tilde{B}_{n,p}(e_0;x)\\ & =3(n+p)^2x^2(1-x)^2+(n+p)\left(x(1-x)-6x^2(1-x)^2\right). \end{align*}
</div>
<p> <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.5</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For any \(x\in [0,1]\) and \(n\in \mathbb {N}\), the following hold: </p>
<div class="displaymath" id="a0000000023">
  \begin{align*}  T^{\ast }_{n,0}\left(\tilde{B}_{n,p};x\right)& =1,\\ T^{\ast }_{n,1}\left(\tilde{B}_{n,p};x\right)& =px,\\ T^{\ast }_{n,2}\left(\tilde{B}_{n,p};x\right)& =(px)^2+(n+p)x(1-x),\\ T^{\ast }_{n,3}\left(\tilde{B}_{n,p};x\right)& =(px)^3+3p(n+p)x^2(1-x)+(n+p)x(1-x)(1-2x),\\ T^{\ast }_{n,4}\left(\tilde{B}_{n,p};x\right)& =(px)^4+6p^2(n+p)x^3(1-x)+4p(n+p)x^2(1-x)(1-2x)\\ & \quad +3(n+p)^2x^2(1-x)^2+(n+p)\left(x(1-x)-6x^2(1-x)^2\right). \end{align*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000024">
  <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="#3.3" class="eqref">3.22</a>, <a href="#3.4" class="eqref">3.22</a> and Lemma <a href="#lem1">3.4</a>, the identities follow. <div class="proof_wrapper" id="a0000000025">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.6</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For any \(x\in [0,1]\), the following relations hold </p>
<div class="equation" id="3.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.5} \lim \limits _{n\to \infty }T^{\ast }_{n,0}\left(\tilde{B}_{n,p};x\right)=1, \end{equation}
  </div>
  <span class="equation_label">3.23</span>
</p>
</div>
<div class="equation" id="3.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.6} \lim \limits _{n\to \infty }\tfrac {T^{\ast }_{n,2}\left(\tilde{B}_{n,p};x\right)}{n}=x(1-x), \end{equation}
  </div>
  <span class="equation_label">3.24</span>
</p>
</div>
<div class="equation" id="3.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.7} \lim \limits _{n\to \infty }\tfrac {T^{\ast }_{n,4}\left(\tilde{B}_{n,p};x\right)}{n^2}=3(x(1-x))^2 \end{equation}
  </div>
  <span class="equation_label">3.25</span>
</p>
</div>
<p> and there exist </p>
<div class="equation" id="3.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.8} T^{\ast }_{n,0}\left(\tilde{B}_{n,p};x\right)=1=k_0, \end{equation}
  </div>
  <span class="equation_label">3.26</span>
</p>
</div>
<div class="equation" id="3.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.9} \tfrac {T^{\ast }_{n,2}\left(\tilde{B}_{n,p};x\right)}{n}\leq \tfrac {1}{4}=k_2, \end{equation}
  </div>
  <span class="equation_label">3.27</span>
</p>
</div>
<div class="equation" id="3.10">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.10} \tfrac {T^{\ast }_{n,4}\left(\tilde{B}_{n,p};x\right)}{n^2}\leq \tfrac {3}{16}=k_4, \end{equation}
  </div>
  <span class="equation_label">3.28</span>
</p>
</div>
<p> for any \(x\in [0,1]\) and \(n\in \mathbb {N}\). </p>

  </div>
</div> <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> The identities <a href="#3.5" class="eqref">3.23</a>–<a href="#3.7" class="eqref">3.25</a> follow immediately from Lemma <a href="#lem2">3.5</a>, while <a href="#3.8" class="eqref">3.26</a>–<a href="#3.10" class="eqref">3.28</a> yield from <a href="#3.5" class="eqref">3.23</a>–<a href="#3.7" class="eqref">3.25</a>. <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000028">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C([0,1+p])\) be a function. If \(x\in [0,1]\) and \(f\) is \(s\) times differentiable in a neighborhood of \(x\), then </p>
<div class="equation" id="3.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.11} \lim \limits _{n\to \infty }\tilde{B}_{n,p}(f;x)=f(x), \end{equation}
  </div>
  <span class="equation_label">3.29</span>
</p>
</div>
<p> for \(s=0;\) </p>
<div class="equation" id="3.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.12} \lim \limits _{n\to \infty }n\left(\tilde{B}_{n,p}(f;x)-f(x)\right)=pxf^{(1)}(x)+\tfrac {x(1-x)}{2}f^{(2)}(x), \end{equation}
  </div>
  <span class="equation_label">3.30</span>
</p>
</div>
<p> for \(s=2;\) </p>
<div class="displaymath" id="3.13">
  \begin{align} \label{3.13} & \lim \limits _{n\to \infty }n^2 \left(\tilde{B}_{n,p}(f;x)-f(x)-\tfrac {px}{n}f^{(1)}(x)-\tfrac {(px)^2+(n+p)x(1-x)}{2n^2}f^{(2)}(x)\right)=\\ & =\tfrac {3px^2(1-x)+x(1-x)(1-2x)}{6}f^{(3)}(x)+\tfrac {(x(1-x))^2}{8}f^{(4)}(x)\nonumber , \end{align}
</div>
<p> for \(s=4\) and </p>
<div class="equation" id="3.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.14} \lim \limits _{n\to \infty }n^{s-\alpha _s}\left(\tilde{B}_{n,p}(f;x)-\sum \limits _{i=0}^s\tfrac {f^{(i)}(x)}{i!\cdot n^i}T^{\ast }_{n,i}\left(\tilde{B}_{n,p};x\right)\right)=0, \end{equation}
  </div>
  <span class="equation_label">3.32</span>
</p>
</div>
<p> for \(s\geq 4.\) </p>
<p>Assume that \(f\) is \(s\) times differentiable on \([0,1+p],\) then the convergence in <a href="#3.11" class="eqref">3.29</a>–<a href="#3.14" class="eqref">3.32</a> is uniform on \([0,1]\subset [0,1+p]\). Moreover, we get </p>
<div class="equation" id="3.15">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.15} \left|\tilde{B}_{n,p}(f;x)-f(x)\right|\leq \tfrac {5}{4}\cdot \omega _1\left(f;\tfrac {1}{\sqrt{n}}\right), \end{equation}
  </div>
  <span class="equation_label">3.33</span>
</p>
</div>
<p> for \(s=0\) and </p>
<div class="displaymath" id="3.16">
  \begin{align} \label{3.16} n\left|\tilde{B}_{n,p}(f;x)\! -\! f(x)\! -\! \tfrac {px}{n}f^{(1)}(x)\! -\! \tfrac {(px)^2+(n+p)x(1-x)}{2n^2}f^{(2)}(x)\right|\leq \tfrac {7}{32}\cdot \omega _1\left(f^{(2)};\tfrac {1}{\sqrt{n}}\right), \end{align}
</div>
<p> for \(s=2.\) </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000029">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> It follows from Theorem <a href="#th1">2.1</a>, with \(\alpha _0=0\), \(\alpha _2=1\) and \(\alpha _4=2\), taking into account Lemma <a href="#lem2">3.5</a> and Lemma <a href="#lem3">3.6</a>. <div class="proof_wrapper" id="a0000000030">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000031">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.7</span>
  </div>
  <div class="remark_thmcontent">
  <p>The above theorem, by the relation <a href="#3.14" class="eqref">3.32</a> generalizes the asymptotic behavior of the Schurer operators and in the particular case \(s=2\) we recover formula <a href="#3.12" class="eqref">3.30</a>, for twice continuously differentiable function, proved first by D. Bărbosu <span class="cite">
	[
	<a href="#DB1" >5</a>
	]
</span>. We get also the asymptotic behavior in the particular case \(s=4\) and quantitative forms in terms of the modulus of continuity, for the same operators. Various quantitative forms of Voronovskaja’s 1932 result <span class="cite">
	[
	<a href="#EVV" >22</a>
	]
</span> dealing with the asymptotic behavior of the Bernstein type operators are discussed also in several recent papers <span class="cite">
	[
	<a href="#HHGPPIR" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#HHG" >11</a>
	]
</span> and <span class="cite">
	[
	<a href="#HHGGT" >12</a>
	]
</span>, where better estimate close to the endpoints \(0\) and \(1\) then the global one was established.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h2 id="a0000000032">3.2 Schurer-Stancu operators</h2>
<p> Using the same construction form preliminaries, we assume that \(I=[0,1+p]\), \(J=[0,1]\), \(E(I)=C([0,1+p])\), \(F(J)=C([0,1])\), the role of \(n\) is played by \(n+p\). Then the functions \(\varphi _{n+p,k}:[0,1]\to \mathbb {R}\) are defined by \(\varphi _{n+p,k}(x):=\tilde{p}_{n,k}(x)\), for any \(x\in [0,1]\), \(n,k\in \mathbb {N}_0\), \(n\neq 0\) and the functionals \(A^{(\alpha ,\beta )}_{n+p,k}:C([0,1+p])\to \mathbb {R}\) are defined by </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  A^{(\alpha ,\beta )}_{n+p,k}(f):=f\left(\tfrac {k+\alpha }{n+\beta }\right), {\rm for\  any}\  n,k\in \mathbb {N}_0, n\neq 0. \end{equation*}
</div>
<p> In this case one obtains the Schurer-Stancu operators, with </p>
<div class="displaymath" id="3.17">
  \begin{align} \label{3.17} T_{n,i}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& =n^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)A_{n+p,k}^{(\alpha ,\beta )}\left(\psi _x^i\right)=n^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)\left(\tfrac {k+\alpha }{n+\beta }-x\right)^i\\ & \nonumber =\left(\tfrac {n}{n+\beta }\right)^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)\left(k-(n+p)x+\alpha +px-\beta x\right)^i\\ & \nonumber =\left(\tfrac {n}{n+\beta }\right)^i\sum \limits _{k=0}^{n+p}\tilde{p}_{n,k}(x)\sum \limits _{l=0}^i\tbinom {i}{l}(k-(n+p))^l(\alpha +px-\beta x)^{i-l}\\ & \nonumber =\left(\tfrac {n}{n+\beta }\right)^i\sum \limits _{l=0}^i\tbinom {i}{l}(\alpha +px-\beta x)^{i-l}T_{n+p,l}(x), \end{align}
</div>
<p> where \(T_{n+p,l}(x)\) were given in <a href="#3.4" class="eqref">3.22</a>. </p>
<p><div class="lemma_thmwrapper " id="lem4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.8</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For any \(x\in [0,1]\) and \(n\in \mathbb {N}\), the following hold: </p>
<div class="displaymath" id="a0000000034">
  \begin{align*}  T_{n,0}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& =1,\\ T_{n,1}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& =\tfrac {n}{n+\beta }(\alpha +px-\beta x),\\ T_{n,2}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& =\left(\tfrac {n}{n+\beta }\right)^2\left((\alpha +px-\beta x)^2+(n+p)x(1-x)\right),\\ T_{n,3}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& =\left(\tfrac {n}{n+\beta }\right)^3\left((\alpha +px-\beta x)^3\right.\\ & \quad \left.+3(\alpha +px-\beta x)(n+p)x(1-x)\! +\! (n\! +\! p)x(1\! -\! x)(1\! -\! 2x)\right),\\ T_{n,4}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)& = \left(\tfrac {n}{n+\beta }\right)^4\left((\alpha +px-\beta x)^4\right.\\ & \quad \left.+6(\alpha +px-\beta x)^2(n+p)x(1-x)\right.\\ & \quad \left.+4(\alpha +px-\beta x)(n+p)x(1-x)(1-2x)\right.\\ & \quad \left.+3(n\! +\! p)^2(x(1-x))^2\! +\! (n+p)\left(x(1-x)\! -\! 6(x(1-x))^2\right)\right). \end{align*}
</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> Using <a href="#3.17" class="eqref">3.35</a> and Lemma <a href="#lem1">3.4</a>, it follows the identities. <div class="proof_wrapper" id="a0000000036">
  <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="lemma_thmwrapper " id="lem5">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.9</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For any \(x\in [0,1]\), the following relations hold </p>
<div class="equation" id="3.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.18} \lim \limits _{n\to \infty }T_{n,0}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)=1, \end{equation}
  </div>
  <span class="equation_label">3.36</span>
</p>
</div>
<div class="equation" id="3.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.19} \lim \limits _{n\to \infty }\tfrac {T_{n,2}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)}{n}=x(1-x), \end{equation}
  </div>
  <span class="equation_label">3.37</span>
</p>
</div>
<div class="equation" id="3.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.20} \lim \limits _{n\to \infty }\tfrac {T_{n,4}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)}{n^2}=3(x(1-x))^2, \end{equation}
  </div>
  <span class="equation_label">3.38</span>
</p>
</div>
<p> and there exist </p>
<div class="equation" id="3.21">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.21} T_{n,0}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)=1=k_0, \end{equation}
  </div>
  <span class="equation_label">3.39</span>
</p>
</div>
<div class="equation" id="3.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.22} \tfrac {T_{n,2}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)}{n}\leq \tfrac {1}{4}=k_2, \end{equation}
  </div>
  <span class="equation_label">3.40</span>
</p>
</div>
<div class="equation" id="3.23">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.23} \tfrac {T_{n,4}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)}{n^2}\leq \tfrac {3}{16}=k_4, \end{equation}
  </div>
  <span class="equation_label">3.41</span>
</p>
</div>
<p> for any \(x\in [0,1]\) and \(n\in \mathbb {N}\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000037">
  <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 identities <a href="#3.18" class="eqref">3.36</a>–<a href="#3.20" class="eqref">3.38</a> follow immediately from Lemma <a href="#lem4">3.8</a>, while <a href="#3.21" class="eqref">3.39</a>–<a href="#3.23" class="eqref">3.41</a> yield from <a href="#3.18" class="eqref">3.36</a>–<a href="#3.20" class="eqref">3.38</a>. <div class="proof_wrapper" id="a0000000038">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000039">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C([0,1+p])\) be a function. If \(x\in [0,1]\) and \(f\) is \(s\) times differentiable in a neighborhood of \(x\), then </p>
<div class="equation" id="3.24">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.24} \lim \limits _{n\to \infty }\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)=f(x), \end{equation}
  </div>
  <span class="equation_label">3.42</span>
</p>
</div>
<p> for \(s=0;\) </p>
<div class="equation" id="3.25">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.25} \lim \limits _{n\to \infty }n\left(\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)-f(x)\right)=(\alpha +px-\beta x)f^{(1)}(x)+\tfrac {x(1-x)}{2}f^{(2)}(x), \end{equation}
  </div>
  <span class="equation_label">3.43</span>
</p>
</div>
<p> for \(s=2;\) </p>
<div class="displaymath" id="3.26">
  \begin{align} \label{3.26} & \lim \limits _{n\to \infty }n^2\left(\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)-f(x)-\tfrac {\alpha +px-\beta x}{n+\beta }f^{(1)}(x)\right. \\ & \quad \quad \quad \quad \nonumber \left.-\tfrac {(\alpha +px-\beta x)^2+(n+p)x(1-x)}{2(n+\beta )^2}f^{(2)}(x)\right)= \\ & \nonumber =\tfrac {3(\alpha +px-\beta x)x(1-x)+x(1-x)(1-2x)}{6}f^{(3)}(x)+\tfrac {(x(1-x))^2}{8}f^{(4)}(x)\nonumber , \end{align}
</div>
<p> for \(s=4\) and </p>
<div class="equation" id="3.27">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.27} \lim \limits _{n\to \infty }n^{s-\alpha _{s}}\left(\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)-\sum \limits _{i=0}^s\tfrac {f^{(i)}(x)}{i!\cdot n^i}T_{n,i}^{\ast }\left(\tilde{S}_{n,p}^{(\alpha ,\beta )};x\right)\right)=0, \end{equation}
  </div>
  <span class="equation_label">3.44</span>
</p>
</div>
<p> for \(s\geq 4.\) Assume that \(f\) is \(s\) times differentiable on \([0,1+p]\), then the convergence from <a href="#3.24" class="eqref">3.42</a>–<a href="#3.27" class="eqref">3.44</a> is uniform on \([0,1]\subset [0,1+p]\). Moreover, we get </p>
<div class="equation" id="3.28">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.28} \left|\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)-f(x)\right|\leq \tfrac {5}{4}\cdot \omega _1\left(f;\tfrac {1}{\sqrt{n}}\right), \end{equation}
  </div>
  <span class="equation_label">3.45</span>
</p>
</div>
<p> for \(s=0\) and </p>
<div class="displaymath" id="3.29">
  \begin{align} \label{3.29} & n\left|\tilde{S}_{n,p}^{(\alpha ,\beta )}(f;x)-f(x)-\tfrac {\alpha +px-\beta x}{n+\beta }f^{(1)}(x)-\tfrac {(\alpha +px-\beta x)^2+(n+p)^2x(1-x)}{2(n+\beta )^2}f^{(2)}(x)\right|\leq \\ & \nonumber \leq \tfrac {7}{32}\cdot \omega _1\left(f^{(2)};\tfrac {1}{\sqrt{n}}\right)\nonumber , \end{align}
</div>
<p> for \(s=2.\) </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000040">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> It follows from Theorem <a href="#th1">2.1</a>, with \(\alpha _0=0,\) \(\alpha _2=1\) and \(\alpha _4=2\), taking into account Lemma <a href="#lem4">3.8</a> and Lemma <a href="#lem5">3.9</a>. <div class="proof_wrapper" id="a0000000041">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000042">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.10</span>
  </div>
  <div class="remark_thmcontent">
  <p>The above theorem, by the relation <a href="#3.27" class="eqref">3.44</a> and by some particular cases given at <a href="#3.25" class="eqref">3.43</a>, respectively <a href="#3.26" class="eqref">3.44</a> generalizes the asymptotic behavior of the Schurer-Stancu operators. Concerning quantitative forms in terms of the modulus of continuity, it is easily to remark that, for the Schurer-Stancu operators we get estimates as good as in the case of the Schurer operators.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
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</dd>
  <dt><a name="OA">2</a></dt>
  <dd><p><i class="sc">Agratini, O.</i>, <em>Approximation by linear operators</em>, Presa Universitară Clujeană, Cluj-Napoca, 2000 (in Romanian). </p>
</dd>
  <dt><a name="FAMC">3</a></dt>
  <dd><p><i class="sc">Altomare, F.</i> and <i class="sc">Campiti, M.</i>, <em>Korovkin-type Approximation Theory and its Applications</em>, de Gruyter Series Studies in Mathematics, Walter de Gruyter &amp; Co. Berlin, <b class="bf">17</b>, New York, 1994. </p>
</dd>
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  <dd><p><i class="sc">Bărbosu, D.</i> and <i class="sc">Bărbosu, M.</i>, <em>Some properties of the fundamental polynomials of Bernstein-Schurer operators</em>, Buletin Şt. Univ. Baia Mare, <b class="bf">XVIII</b>, no. 2, pp.&#160;133–136, 2002. </p>
</dd>
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  <dd><p><i class="sc">Bărbosu, D.</i>, <em>Voronovskaja theorem for Bernstein-Schurer operators</em>, Buletin Şt. Univ. Baia Mare, <b class="bf">XVIII</b>, no. 2, pp.&#160;137–140, 2002. </p>
</dd>
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</dd>
  <dt><a name="DB3">7</a></dt>
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</dd>
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</dd>
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</dd>
  <dt><a name="HHGPPIR">10</a></dt>
  <dd><p><i class="sc">Gonska, H. H., Piţul, P.</i> and <i class="sc">Raşa, I.</i>, <em>On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators</em>, In “Numerical Analysis and Approximation Theory" (Proc. Int. Conf. Cluj-Napoca 2006, ed. by O. Agratini and P. Blaga), Casa Cărţii de Ştiinţă, pp.&#160;55–80, 2006. </p>
</dd>
  <dt><a name="HHG">11</a></dt>
  <dd><p><i class="sc">Gonska, H. H.</i>, <em>On the degree of approximation in Voronovskaja’s theorem</em>, Studia Univ. Babeş Bolyai, Math., <b class="bf">LII</b> (2007), no. 3, pp.&#160;103–115 </p>
</dd>
  <dt><a name="HHGGT">12</a></dt>
  <dd><p><i class="sc">Gonska, H. H.</i> and <i class="sc">Tachev, G.</i>, <em>A quantitative variant of Voronovskaja’s theorem</em>, Results in Math., <b class="bf">53</b>, pp.&#160;287–294, 2009. </p>
</dd>
  <dt><a name="SKZZ">13</a></dt>
  <dd><p><i class="sc">Karlin, S.</i> and <i class="sc">Ziegler, Z.</i>, <em>Iteration of positive approximation operators</em>, J. Approx. Theory, <b class="bf">3</b>, pp.&#160;310–339, 1970. </p>
</dd>
  <dt><a name="DMPIB">14</a></dt>
  <dd><p><i class="sc">Miclăuş, D.</i> and <i class="sc">Braica, P. I.</i>, <em>The generalization of some results for Bernstein and Stancu operators</em>, Creat. Math. Inf., <b class="bf">20</b>, no. 2, pp. 147–156, 2011. </p>
</dd>
  <dt><a name="DMPIB1">15</a></dt>
  <dd><p><i class="sc">Miclăuş, D.</i> and <i class="sc">Braica, P. I.</i>, <em>Some results concerning calculation of the test functions by Bernstein type operators</em>, Acta Univ. Apulensis, <b class="bf">28</b>, pp. 135–142, 2011. </p>
</dd>
  <dt><a name="OTPMF">16</a></dt>
  <dd><p><i class="sc">Pop, O. T.</i> and <i class="sc">Farcaş, M.</i>, <em>About Bernstein polynomial and the Stirling’s numbers of second type</em>, Creative Math. &amp; Inf., <b class="bf">14</b>, pp.&#160;53–56, 2005. </p>
</dd>
  <dt><a name="OTP1">17</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no1-art9"> <i class="sc">Pop, O. T.</i>, <em>The generalization of Voronovskaja’s theorem for a class of linear and positive operators</em>, Rev. Anal. Num. Théor. Approx., <b class="bf">34</b>, no. 1, pp.&#160;79–91, 2005. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="OTP2">18</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no2-art6"> <i class="sc">Pop, O. T.</i>, <em>About a general property for a class of linear and positive operators and applications</em>, Rev. Anal. Num. Théor. Approx., <b class="bf">34</b>, no. 2, pp.&#160;175–180, 2005. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="OTPDBPIB">19</a></dt>
  <dd><p><i class="sc">Pop, O. T., Bărbosu, D.</i> and <i class="sc">Braica, P. I.</i>, <em>Some results regarding the Bernstein polynomials</em>, Acta Universitatis Apulensis, <b class="bf">22</b>, pp.&#160;243–247, 2010. </p>
</dd>
  <dt><a name="FS">20</a></dt>
  <dd><p><i class="sc">Schurer, F.</i>, <em>Linear positive operators in approximation theory</em>, Math. Inst. Techn., Univ. Delft Report, 1962. </p>
</dd>
  <dt><a name="JS">21</a></dt>
  <dd><p><i class="sc">Stirling, J.</i>, <em>Differential Method: A Treatise of the Summation and Interpolation of Infinite Series</em>, London, 1730, (in Latin). </p>
</dd>
  <dt><a name="EVV">22</a></dt>
  <dd><p><i class="sc">Voronovskaja, E. V.</i>, <em>Détérmination de la forme asymtotique d’approximation des fonctions par des polinômes de Bernstein</em>, C. R. Acad. Sci. URSS, pp.&#160;79–85, 1932. </p>
</dd>
</dl>


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