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

<body>

<div class="wrapper">

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


<div class="main-text">


<div class="titlepage">
<h1>On Convergence of Chlodovsky Type Durrmeyer Polynomials in Variation Seminorm</h1>
<p class="authors">
<span class="author"> Özlem Öksüzer Yılık\(^{\ast }\), Harun Karsli\(^{\bullet }\) Fatma Tasdelen\(^{\ast }\)</span>
</p>
<p class="date">August 18, 2017. Accepted: February 22, 2018. Published online: August 6, 2018.</p>
</div>
<p>\(^{\ast }\)Department of Mathematics, Faculty of Science, University of Ankara, Ankara, Turkey, e-mail: <span class="ttfamily">o.oksuzer@hotmail.com.tr; tasdelen@science.ankara.edu.tr.</span> </p>
<p>\(^{\bullet }\)Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Bolu, Turkey, e-mail: <span class="ttfamily">karsli_h@ibu.edu.tr</span>. </p>

<div class="abstract"><p> This paper deals with the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm. </p>
<p><b class="bf">MSC.</b> 41A25, 41A35, 41A36. </p>
<p><b class="bf">Keywords.</b> Chlodovsky polynomials, Durrmeyer polynomials, convergence in variation seminorm, rate of convergence, absolutely continuous functions. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(X_{loc}[0,\infty )\) be the class of all complex-valued functions locally bounded on \([0,\infty )\). For \(x\in X_{loc}[0,\infty )\), the Chlodovsky polynomials \(C_{n}f\) are defined as: </p>
<div class="equation" id="chl">
<p>
  <div class="equation_content">
    \begin{equation}  \left( C_{n}f\right) \left( x\right) =\sum \limits _{k=0}^{n}f\left( \tfrac {k}{n}b_{n}\right) \tbinom {n}{k}\left( \tfrac {x}{b_{n}}\right) ^{k}\left( 1-\tfrac {x}{b_{n}}\right) ^{n-k},\text{ }(0\leq x\leq b_{n}) \label{chl} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>where \(n\in \mathbb {N} \) and \(\left( b_{n}\right) \) is an increasing sequence of positive numbers satisfying \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \) and \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}}{n}=0.\) </p>
<p>These polynomials were introduced by I. Chlodovsky <span class="cite">
	[
	<a href="#Chl" >1</a>
	]
</span> in 1937 in generalization of the Bernstein polynomials, the case \(b_{n}=1,\) \(n\in \mathbb {N} \), which approximate the function \(f\) on the interval \(\left[ 0,1\right] \). Some other generalizations of the Bernstein polynomials defined on unbounded sets can be found in <span class="cite">
	[
	<a href="#Hermann" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#Butzer" >3</a>
	]
</span>. Works on Chlodovsky polynomials are fewer, since they are defined on an unbounded interval \([0,\infty ).\) </p>
<p>This generalizes Chlodovsky polynomials by incorporating Durrmeyer operators <span class="cite">
	[
	<a href="#Durr" >4</a>
	]
</span>, hence the name Chlodovsky-Durrmeyer operators</p>
<div class="equation" id="cd">
<p>
  <div class="equation_content">
    \begin{equation}  \left( D_{n}f\right) \left( x\right) =\tfrac {n+1}{b_{n}}\sum \limits _{k=0}^{n}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt,\text{ \  }0\leq x\leq b_{n} \label{cd} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>where \(\left( b_{n}\right) \) is a positive increasing sequence with the properties \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \), \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}}{n}=0\) and \(\  p_{n,k}\left( x\right) :=\binom {n}{k}x^{k}\left( 1-x\right) ^{n-k}\) is the Bernstein basis. We may also mention that some articles related to Chlodovsky-Durrmeyer operators and their different generalizations are given in <span class="cite">
	[
	<a href="#kt" >5</a>
	]
</span>–<span class="cite">
	[
	<a href="#Kar" >6</a>
	]
</span>. </p>
<p>The main motivation for this paper is to study the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm. The first research devoted to the variation detracting property and the convergence in variation of a sequence of linear positive operators was due to Lorentz <span class="cite">
	[
	<a href="#Lorentz" >7</a>
	]
</span>. Later in <span class="cite">
	[
	<a href="#BB" >8</a>
	]
</span>, authors have introduced, developed in details and studied the deep interconnections between variation detracting property and the convergence in variation for Bernstein-type polynomials and singular convolution integrals. After this fundamental study, the convergence in variation seminorm has become a new research field in the theory of approximation. For further reading on different operators, we refer to readers to <span class="cite">
	[
	<a href="#Agratini" >9</a>
	]
</span>–<span class="cite">
	[
	<a href="#OKT" >15</a>
	]
</span>. </p>
<h1 id="a0000000003">2 Notation and Auxiliary Results</h1>
<p>For the notation; let \(I\subset \mathbb {R} \) be a bounded or unbounded interval. We denote by \(V_{\left[ I\right] }\left[ f\right] \) the total Jordan variation of the function \(f:I\rightarrow \mathbb {R} \). We deal with the class \(BV\left( I\right) \) of all the functions of bounded variation on \(I\subset \mathbb {R} \), endowed with norm \(\left\Vert .\right\Vert _{BV\left( I\right) }\), where </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \left\Vert f\right\Vert _{BV\left( I\right) }:=V_{\left[ I\right] }\left[ f\right] +\left\vert f\left( a\right) \right\vert ,\text{ \  }f\in BV(I), \end{equation*}
</div>
<p>\(a\  \)being any fixed point belonging to the interval \(I.\) If we remove the term \(\left\vert f\left( a\right) \right\vert ,\) \(V_{\left[ I\right] }\left[ f\right] \) turns into a seminorm, say \(\left\vert .\right\vert _{BV\left( I\right) }\) on the same space. So we shall say, \(TV(I)\) of all the functions of bounded variation on \(I\subset \mathbb {R} \), endowed with seminorm</p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  \left\Vert f\right\Vert _{BV\left( I\right) }:=V_{\left[ I\right] }\left[ f\right] . \end{equation*}
</div>
<p>Some interesting properties of the space \(TV(I)\) are presented in <span class="cite">
	[
	<a href="#BB" >8</a>
	]
</span>. </p>
<p>In order to obtain a convergence result in the variation seminorm, it is necessary and important to state the variation detracting property. Let \(L\) be a linear operator acting on a given space \(S\) of real-valued functions defined on \(I\) such that \(BV\left( I\right) \subset S.\) The operator \(L\) possesses the variation detracting property if</p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  V_{\left[ I\right] }\left[ Lf\right] \leq V_{\left[ I\right] }\left[ f\right] \text{ , }f\in BV(I)\text{,} \end{equation*}
</div>
<p>holds, <i class="itshape">i.e.</i> positive linear operators from the space of functions of bounded variation into itself do not increase the total variation of functions. </p>
<p>\(AC\left( I\right) \) stands for the space of all absolutely continuous real-valued functions defined on \(I\) is a closed subspace of \(TV\left( I\right) \) with respect to the convergence induced by the seminorm \(\left\Vert f\right\Vert _{TV\left( I\right) }\). Moreover, if \(\underset {n\rightarrow \infty }{\lim }V_{I}\left[ g_{n}-f\right] =0\) for a sequence \(\left( g_{n}\right) _{n\geq 1}\), \(g_{n}\in AC\left( I\right) \), \(n\in \mathbb {N} ,\) then also \(f\in AC\left[ 0,1\right] \) and</p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  V_{I}\left[ g_{n}-f\right] =\int \limits _{I}\left\vert g_{n}^{\prime }\left( t\right) -f^{\prime }\left( t\right) \right\vert dt=\left\Vert g_{n}^{\prime }-f^{\prime }\right\Vert , \end{equation*}
</div>
<p>where</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  \left\Vert f\right\Vert :=\left\Vert f\right\Vert _{L_{1}\left( I\right) }. \end{equation*}
</div>
<p>So, convergence in variation of \(\left( g_{n}\right) _{n\geq 1}\subset AC\left( I\right) \) to \(f,\) represents the convergence of the derivatives \(\left( g_{n}^{\prime }\right) _{n\geq 1}\) to \(f^{\prime }\) in the norm \(L_{1}\left( I\right) \), the Banach space of all real-valued Lebesgue integrable functions defined on \(I.\) </p>
<p>Let us define the sum moments as in <span class="cite">
	[
	<a href="#Karsli" >11</a>
	]
</span>:</p>
<div class="equation" id="2">
<p>
  <div class="equation_content">
    \begin{equation}  T_{n,m}\left( x\right) =\sum \limits _{k=0}^{n}\left[ kb_{n}-nx\right] ^{m}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \label{2} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>where \(m\in \mathbb {N} _{0}\) (the set of non-negative integers). Then there hold the following identities (see, e.g., <span class="cite">
	[
	<a href="#Karsli" >11</a>
	]
</span>)</p>
<div class="equation" id="mom">
<p>
  <div class="equation_content">
    \begin{equation}  T_{n,m}\left( x\right) =\left\{  \begin{tabular}{ll} $1$ 

&  $m=0$ 

\\ $0$ 

&  $m=1$ 

\\ $nx\left( b_{n}-x\right) $ 

&  $m=2$ 

\\ $nx\left( b_{n}-x\right) \left( b_{n}-2x\right) $ 

&  $m=3$ 

\\ $nx\left( b_{n}-x\right) \left( b_{n}^{2}+3\left( n-2\right) xb_{n}-3\left( n-2\right) x^{2}\right) $ 

&  $m=4.$

\end{tabular}\right. \label{mom} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>Let us define for the central moments of order \(m\in \mathbb {N} _{0},\)</p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  T_{n,m}^{\ast }\left( x\right) =\sum \limits _{k=0}^{n}\left( \tfrac {kb_{n}}{n}-x\right) ^{m}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \end{equation*}
</div>
<p>and for any fixed \(x\in \lbrack 0,\infty )\)</p>
<div class="equation" id="5">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert T_{n,m}^{\ast }\left( x\right) \right\vert \leq A_{m}(x)\tfrac {x\left( b_{n}-x\right) }{b_{n}}\left( \tfrac {b_{n}}{n}\right) ^{\left[ \left( m+1\right) /2\right] }\text{ \  }\left( n\in \mathbb {N} ,\text{ }n>b_{n}\right) , \label{5} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>where \(A_{m}(x)\) denotes a polynomial in \(x,\  \)of degree \(\left[ m/2\right] -1\), with non-negative coefficients independent of \(n\), and \(\left[ a\right] \) denotes the integral part of \(a\). For the proof see Butzer-Karsli <span class="cite">
	[
	<a href="#BK" >16</a>
	]
</span>. </p>
<p>Since</p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \tfrac {d}{dx}p_{n,k}\left( \tfrac {x}{b_{n}}\right) =\tfrac {\left( kb_{n}-nx\right) }{x(b_{n}-x)}p_{n,k}\left( \tfrac {x}{b_{n}}\right) , \end{equation*}
</div>
<p>we can write the following representations for the first derivative of \(\left( D_{n}f\right) \left( x\right) \);</p>
<div class="equation" id="a">
<p>
  <div class="equation_content">
    \begin{equation}  \left( D_{n}f\right) ^{\prime }\left( x\right) =\tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \label{a} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>and</p>
<div class="equation" id="b">
<p>
  <div class="equation_content">
    \begin{equation}  \left( D_{n}f\right) ^{\prime }\left( x\right) =\tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ p_{n,k+1}\left( \tfrac {t}{b_{n}}\right) -p_{n,k}\left( \tfrac {t}{b_{n}}\right) \right] dt. \label{b} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<h1 id="a0000000011">3 Variation Detracting Property of Chlodovsky-Durrmeyer Operators</h1>
<p>In this section, we prove the variation detracting properties of the Chlodovsky-Durrmeyer Operators. </p>
<p><div class="theorem_thmwrapper " id="a0000000012">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f\in TV\left[ 0,b_{n}\right] \), then</p>
<div class="equation" id="14">
<p>
  <div class="equation_content">
    \begin{equation}  V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] \leq V_{\left[ 0,b_{n}\right] }\left[ f\right] \label{14} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>and </p>
<div class="equation" id="15">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] }\leq \left\Vert f\right\Vert _{BV\left[ 0,b_{n}\right] } \label{15} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>hold true. </p>

  </div>
</div> </p>
<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>For convenience we write the Chlodovsky-Durrmeyer operators as:</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  \left( D_{n}f\right) \left( x\right) =\sum \limits _{k=0}^{n}p_{n,k}\left( \tfrac {x}{b_{n}}\right) F_{k,n} \end{equation*}
</div>
<p>where</p>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  F_{k,n}:=\tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{equation*}
</div>
<p>As in (<a href="#b">7</a>), differentiating (<a href="#cd">2</a>) and putting \(\Delta F_{k,n}=F_{k+1,n}-F_{k,n}\)</p>
<div class="displaymath" id="a0000000016">
  \begin{eqnarray}  \left( D_{n}f\right) ^{\prime }\left( x\right) & =& \sum \limits _{k=0}^{n}p_{n,k}^{\prime }\left( \tfrac {x}{b_{n}}\right) F_{k,n}=\tfrac {1}{b_{n}}\sum \limits _{k=1}^{n}\tbinom {n}{k}k\left( \tfrac {x}{b_{n}}\right) ^{k-1}\left( 1-\tfrac {x}{b_{n}}\right) ^{n-k}F_{k,n} \notag \\ & & -\tfrac {1}{b_{n}}\sum \limits _{k=0}^{n-1}\tbinom {n}{k}\left( \tfrac {x}{b_{n}}\right) ^{k}\left( n-k\right) \left( 1-\tfrac {x}{b_{n}}\right) ^{n-k-1}F_{k,n} \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) F_{k+1,n}-\tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) F_{k,n} \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \left[ F_{k+1,n}-F_{k,n}\right] \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \Delta F_{k,n}. \label{3} \end{eqnarray}
</div>
<p>Considering the representation (<a href="#3">10</a>) of<b class="bfseries"> </b>\(\left( D_{n}f\right) ^{\prime }\), one has</p>
<div class="displaymath" id="a0000000017">
  \begin{eqnarray*}  \left\Vert D_{n}f\right\Vert _{TV\left[ 0,b_{n}\right] } & =& V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] =\int \limits _{0}^{b_{n}}\left\vert \left( D_{n}f\right) ^{\prime }\left( x\right) \right\vert dx \\ & \leq & \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert \int \limits _{0}^{b_{n}}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) dx. \end{eqnarray*}
</div>
<p>Since \(\tfrac {n}{b_{n}}\displaystyle \int _{0}^{b_{n}}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) dx=1\), we get</p>
<div class="equation" id="*">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert D_{n}f\right\Vert _{TV\left[ 0,b_{n}\right] }\leq \sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert . \label{*} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>Now,</p>
<div class="displaymath" id="a0000000018">
  \begin{eqnarray*}  \Delta F_{k,n} & =& \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ p_{n,k+1}\left( \tfrac {t}{b_{n}}\right) -p_{n,k}\left( \tfrac {t}{b_{n}}\right) \right] dt \\ & =& \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \Delta p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{eqnarray*}
</div>
<p>Since \(\Delta p_{n,k}=-\frac{b_{n}}{n+1}p_{n+1,k+1}^{\prime }\),</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  p_{n,k}^{\prime }\left( \tfrac {t}{b_{n}}\right) =-\tfrac {n}{b_{n}}\Delta p_{n-1,k-1}\left( \tfrac {t}{b_{n}}\right) , \end{equation*}
</div>
<p>and so we get</p>
<div class="equation" id="**">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \Delta F_{k,n}\right\vert =\left\vert \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ -\tfrac {b_{n}}{n+1}p_{n+1,k+1}^{\prime }\left( \tfrac {t}{b_{n}}\right) \right] dt\right\vert . \label{**} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>From (<a href="#*">11</a>) and (<a href="#**">12</a>), we obtain</p>
<div class="displaymath" id="a0000000020">
  \begin{eqnarray*}  V_{\left[ 0,1\right] }\left[ D_{n}f\right] & =& \sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert =\sum \limits _{k=0}^{n-1}\left\vert -\int \limits _{0}^{b_{n}}f\left( t\right) p_{n+1,k+1}^{\prime }\left( \tfrac {t}{b_{n}}\right) dt\right\vert \\ & =& \sum \limits _{k=0}^{n-1}\left\vert \int \limits _{0}^{b_{n}}f^{\prime }\left( t\right) p_{n+1,k+1}\left( \tfrac {t}{b_{n}}\right) dt\right\vert \\ & \leq & \sum \limits _{k=0}^{n-1}\int \limits _{0}^{b_{n}}p_{n+1,k+1}\left( \tfrac {t}{b_{n}}\right) \left\vert f^{\prime }\left( t\right) \right\vert dt \\ & =& \int \limits _{0}^{b_{n}}\sum \limits _{k=0}^{n-1}\tbinom {n+1}{k+1}\left( \tfrac {t}{b_{n}}\right) ^{k+1}\left( 1-\tfrac {t}{b_{n}}\right) ^{n-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & =& \int \limits _{0}^{b_{n}}\sum \limits _{k=1}^{n}\tbinom {n+1}{k}\left( \tfrac {t}{b_{n}}\right) ^{k}\left( 1-\tfrac {t}{b_{n}}\right) ^{n+1-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & \leq & \int \limits _{0}^{b_{n}}\sum \limits _{k=0}^{n+1}\tbinom {n+1}{k}\left( \tfrac {t}{b_{n}}\right) ^{k}\left( 1-\tfrac {t}{b_{n}}\right) ^{n+1-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000021">
  \begin{eqnarray*} & =& \int \limits _{0}^{b_{n}}\left( \tfrac {t}{b_{n}}+1-\tfrac {t}{b_{n}}\right) ^{n+1}\left\vert f^{\prime }\left( t\right) \right\vert dt=\int \limits _{0}^{b_{n}}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & \leq & V_{\left[ 0,b_{n}\right] }\left[ f\right] . \end{eqnarray*}
</div>
<p>The desired estimate (<a href="#14">8</a>) is now obvious. </p>
<p>Since </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \left( D_{n}f\right) \left( 0\right) =\tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \left\Vert f\right\Vert _{BV\left[ I\right] }:=V_{\left[ I\right] }\left[ f\right] +\left\vert f\left( 0\right) \right\vert , \end{equation*}
</div>
<p>relation (<a href="#15">9</a>) is a result of (<a href="#14">8</a>). Indeed,</p>
<div class="displaymath" id="a0000000024">
  \begin{eqnarray*}  \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] } & =& V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] +\left\vert \left( D_{n}f\right) \left( 0\right) \right\vert \\ & \leq & V_{\left[ 0,b_{n}\right] }\left[ f\right] +\left\vert \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt\right\vert . \end{eqnarray*}
</div>
<p>Since \(f\in TV\left[ 0,b_{n}\right] \) and </p>
<div class="displaymath" id="a0000000025">
  \[  \left\vert \tfrac {n+1}{b_{n}}\int _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt\right\vert =\left\vert f\left( 0\right) \right\vert \leq \left\vert f\left( a\right) \right\vert  \]
</div>
<p> where \(a\) is any fixed point of \(\left[ 0,b_{n}\right] \), we get</p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] }\leq \left\Vert f\right\Vert _{BV\left[ 0,b_{n}\right] }. \end{equation*}
</div>
<p>Thus, the proof of the theorem is complete. <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>
<h1 id="a0000000028">4 Rate of Approximation in \(TV\)-norm</h1>
<p>This section deals with the rates of approximation \(D_{n}g\) to \(g\) in the variation seminorm. </p>
<p>In order to obtain a convergence result in variation seminorm, we assume that \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \) and \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}^{3}}{n}=0\) </p>
<p><div class="theorem_thmwrapper " id="a0000000029">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(g^{\prime \prime }\in AC\left[ 0,b_{n}\right] \), then</p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  V_{\left[ 0,b_{n}\right] }\left[ D_{n}g-g\right] \leq \tfrac {B}{\delta ^{2}}\tfrac {b_{n}^{3}}{n}\left\{  V_{\left[ 0,b_{n}\right] }\left[ g\right] +V_{\left[ 0,b_{n}\right] }\left[ g^{\prime \prime }\right] \right\}  \end{equation*}
</div>
<p>holds true, where \(\delta \) is sufficiently small positive real constant and a constant \(B{\gt}1\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000031">
  <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 Taylor’s formula with integral remainder term, one has</p>
<div class="displaymath" id="4">
  \begin{align}  g\left( \tfrac {k}{n}b_{n}\right) = &  \label{4} g\left( x\right) +\left( \tfrac {k}{n}b_{n}-x\right) g^{\prime }\left( x\right) \\ &  +\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\tfrac {g^{\prime \prime }\left( x\right) }{2}+\tfrac {1}{2}\int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv. \nonumber \end{align}
</div>
<p>From (<a href="#4">13</a>) we obtain</p>
<div class="displaymath" id="a0000000032">
  \begin{align*} & \left( D_{n}g\right) ^{\prime }\left( x\right) = \\ =&  \tfrac {n+1}{b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}g\left( \tfrac {k}{n}b_{n}\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ =& \tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}g\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}g^{\prime }\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\tfrac {\left( n+1\right) }{2b_{n}x(b_{n}-x)}g^{\prime \prime }\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\left( R_{n}g\right) \left( x\right) . \end{align*}
</div>
<p>where</p>
<div class="displaymath" id="6">
  \begin{align} & \left( R_{n}g\right) \left( x\right) = \label{6} \\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}\! -\!  nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right)\!  \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \! \tfrac {k}{n}b_{n}\! -\! v\right) ^{2}\!  g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \nonumber \end{align}
</div>
<p>Calculating (<a href="#2">3</a>) and (<a href="#mom">4</a>), we obtain </p>
<div class="equation" id="11">
<p>
  <div class="equation_content">
    \begin{equation}  \left( D_{n}g\right) ^{\prime }\left( x\right) =g^{\prime }\left( x\right) +\tfrac {b_{n}-2x}{2n}g^{\prime \prime }\left( x\right) +\left( R_{n}g\right) \left( x\right) . \label{11} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>As in the proof of <span class="cite">
	[
	<a href="#OKT" >15</a>
	]
</span>, we choose \(\delta \) a sufficiently small positive real number, let’s divide \(\left( R_{n}g\right) \left( x\right) \) in to two parts as follows;</p>
<div class="equation" id="12">
<p>
  <div class="equation_content">
    \begin{equation}  \left( R_{n}g\right) \left( x\right) =\left( R_{n,1}g\right) \left( x\right) +\left( R_{n,2}g\right) \left( x\right) , \label{12} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>where</p>
<div class="displaymath" id="a0000000033">
  \begin{align} & \left( R_{n,1}g\right) \left( x\right) =\\ \label{12.1} & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \nonumber \\ &  \quad \cdot \sum \limits _{\left\vert \frac{k}{n}-x\right\vert \leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\nonumber \end{align}
</div>
<p>and</p>
<div class="displaymath" id="a0000000034">
  \begin{align} & \left( R_{n,2}g\right) \left( x\right)=\\ \label{12.2} & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \nonumber \\ & \quad \cdot \sum \limits _{\left\vert \frac{k}{n}-x\right\vert {\gt}\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \nonumber \end{align}
</div>
<p>In order to estimate the integration domain of the double integral in the remainder term (<a href="#6">14</a>), we divide the summation into different sums as following;</p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  \left( R_{n,1}g\right) \left( x\right) =A_{2,n}g+A_{5,n}g\text{ ~ and }\left( R_{n,2}g\right) \left( x\right) =A_{1,n}g+A_{3,n}g+A_{4,n}g+A_{6,n}g. \end{equation*}
</div>
<p>Here \(A_{i,n}g\) for \(i=1,\ldots ,6,\) </p>
<div class="displaymath" id="a0000000036">
  \begin{align*} & A_{1,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \int \limits _{0}^{\frac{k}{n}b_{n}}\! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \\ & A_{2,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}
</div>
<div class="displaymath" id="a0000000037">
  \begin{align*} & A_{3,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}
</div>
<div class="displaymath" id="a0000000038">
  \begin{align*} & A_{4,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{x}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \\ & A_{5,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0{\lt}\frac{k}{n}b_{n}-x\leq \delta }\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000039">
  \begin{align*} & A_{6,n}g=\\ & =\frac{n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \! \! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \! \int \limits _{\tfrac {k}{n}b_{n}}^{b_{n}}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}\! -\! v\right) ^{2}\! \! g^{\prime \prime \prime }\left( v\right) dv\right]\! \!  p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{align*}
</div>
<p>It is easy to see that, \(A_{1,n}g+A_{2,n}g=-A_{4,n}g\) and \(A_{5,n}g+A_{6,n}g=-A_{3,n}g.\) So one has</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  \left\vert A_{1,n}g+A_{2,n}g\right\vert =\left\vert -A_{4,n}g\right\vert \leq \left\vert A_{1,n}g\right\vert +\left\vert A_{2,n}g\right\vert \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  \left\vert A_{5,n}g+A_{6,n}g\right\vert =\left\vert -A_{3,n}g\right\vert \leq \left\vert A_{5,n}g\right\vert +\left\vert A_{6,n}g\right\vert . \end{equation*}
</div>
<p>So, we get</p>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  \left\vert \left( R_{n}g\right) \left( x\right) \right\vert \leq 2\left( \left\vert A_{1,n}g\right\vert +\left\vert A_{2,n}g\right\vert +\left\vert A_{5,n}g\right\vert +\left\vert A_{6,n}g\right\vert \right) \end{equation*}
</div>
<p>or</p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \left\vert \left( R_{n}g\right) \left( x\right) \right\vert \leq 2\left( \left\vert -A_{3,n}g\right\vert +\left\vert -A_{4,n}g\right\vert \right) . \end{equation*}
</div>
<p>Now we only estimate \(A_{i,n}g\) for \(i=1,2,5\), and \(6\) respectively. Firstly, let us estimate \(A_{1,n}g\) as follows;</p>
<div class="displaymath" id="a0000000044">
  \begin{align*} & \left\vert A_{1,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \! \int \limits _{0}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}\! g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left(\tfrac {t}{b_{n}}\right)\! \! \  dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left(\tfrac {t}{b_{n}}\right)\! \! \  dt \end{align*}
</div>
<div class="displaymath" id="a0000000045">
  \begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) x^{2}\int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) x^{2}\int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left( n+1\right) x^{2}}{2b_{n}x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & =\tfrac {x^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \end{align*}
</div>
<p>Since</p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) =\sum \limits _{\frac{k}{n}b_{n}-x{\lt}-\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \text{,} \end{equation*}
</div>
<p>we get</p>
<div class="equation" id="7">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert A_{1,n}g\right\vert \leq \tfrac {x^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\frac{k}{n}b_{n}-x<-\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \label{7} \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>Analogously, \(A_{2,n}g\) can be estimated by</p>
<div class="displaymath" id="a0000000047">
  \begin{align*} & \left\vert A_{2,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \end{align*}
</div>
<div class="displaymath" id="a0000000048">
  \begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}^{1/2}\left( \tfrac {x}{b_{n}}\right) \left( kb_{n}-nx\right) ^{2}p_{n,k}^{1/2}\left( \tfrac {x}{b_{n}}\right) . \end{align*}
</div>
<p>In view of Hölder inequality, (<a href="#mom">4</a>) and (<a href="#5">5</a>), we get</p>
<div class="displaymath" id="a0000000049">
  \begin{align*} & \left\vert A_{2,n}g\right\vert \leq \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\left( \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) ^{2}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \right) ^\frac {1}{2}\left( \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) ^{4}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \right) ^\frac {1}{2} \\ & \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert \end{align*}
</div>
<p>which implies that </p>
<div class="equation" id="8">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert A_{2,n}g\right\Vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{8} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>As to the term \(A_{5,n}g\), noting (<a href="#mom">4</a>) and (<a href="#5">5</a>), we have</p>
<div class="displaymath" id="a0000000050">
  \begin{align*} & \left\vert A_{5,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0{\lt}\frac{k}{n}b_{n}-x\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\frac{k}{n}b_{n}-x\leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \end{align*}
</div>
<div class="displaymath" id="a0000000051">
  \begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\frac{k}{n}b_{n}-x\leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left\vert kb_{n}-nx\right\vert ^{3}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \end{align*}
</div>
<p>As in the proof of \(A_{2,n}g\), one has</p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  \left\vert A_{5,n}g\right\vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert , \end{equation*}
</div>
<p>which yields</p>
<div class="equation" id="9">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert A_{5,n}g\right\Vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{9} \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>Finally to the next term \(A_{6,n}g\), we have </p>
<div class="displaymath" id="a0000000053">
  \begin{align*} & \left\vert A_{6,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right)\! \! \!  \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\notag \end{align*}
</div>
<div class="displaymath" id="a0000000054">
  \begin{align} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \! \! \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left\vert \! \! \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\! \! \left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \end{align}
</div>
<div class="displaymath" id="a0000000055">
  \begin{align} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \\ & \leq \tfrac {\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \label{10} \end{align}
</div>
<p>Collecting (<a href="#7">19</a>) and (<a href="#10">21</a>), we obtain</p>
<div class="displaymath" id="a0000000056">
  \begin{eqnarray}  \left\vert A_{1,n}g\right\vert +\left\vert A_{6,n}g\right\vert & \leq & \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \notag \\ & \leq & \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert n\! \! \sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( \tfrac {k}{n}b_{n}\! -\! x\right) \! \tfrac {\left( \frac{k}{n}b_{n}\! -x\right) ^{2}}{\delta ^{2}}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \notag \\ & =& \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \frac{n}{\delta ^{2}}\sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( \tfrac {k}{n}b_{n}-x\right) ^{3}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \\ & \leq & \tfrac {b_{n}^{3}}{n\delta ^{2}}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{16} \end{eqnarray}
</div>
<p>In view of (<a href="#16">23</a>), one obtains</p>
<div class="equation" id="17">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert A_{1,n}g\right\Vert +\left\Vert A_{6,n}g\right\Vert \leq \tfrac {b_{n}^{3}}{n\delta ^{2}}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{17} \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>Thus, altogether with the results in (<a href="#8">20</a>), (<a href="#9">21</a>) and (<a href="#17">24</a>), we have for (<a href="#12">16</a>)</p>
<div class="displaymath" id="a0000000057">
  \begin{equation*}  \left\Vert R_{n}g\right\Vert \leq 2\left( \tfrac {\sqrt{b_{n}}}{n}+\tfrac {\sqrt{b_{n}}}{n}+\tfrac {b_{n}^{3}}{n\delta ^{2}}\right) \left\Vert g^{\prime \prime \prime }\right\Vert \leq \tfrac {6b_{n}^{3}}{\delta ^{2}n}\left\Vert g^{\prime \prime \prime }\right\Vert . \end{equation*}
</div>
<p>Finally we obtain by using (<a href="#11">15</a>)</p>
<div class="displaymath" id="a0000000058">
  \begin{equation*}  \left\Vert \left( D_{n}g\right) ^{\prime }-g^{\prime }\right\Vert \leq \tfrac {b_{n}}{2n}\left\Vert g^{\prime \prime }\right\Vert +\tfrac {6b_{n}^{3}}{\delta ^{2}n}\left\Vert g^{\prime \prime \prime }\right\Vert . \end{equation*}
</div>
<p>According to Stein’s inequality (see, e.g., <span class="cite">
	[
	<a href="#RM" >17</a>
	, 
	Th.
	
	A10.1
	]
</span>) one has</p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*}  \left\Vert g^{\prime \prime }\left( x\right) \right\Vert _{L_{1}\left( 0,1\right) } & \leq & C\sqrt{\left\Vert g^{\prime }\left( x\right) \right\Vert _{L_{1}\left( 0,1\right) }\left\Vert g^{\prime \prime \prime }\right\Vert _{L_{1}\left( 0,1\right) }} \\ & \leq & C\left( \left\Vert g^{\prime }\right\Vert _{L_{1}\left( 0,1\right) }+\left\Vert g^{\prime \prime \prime }\right\Vert _{L_{1}\left( 0,1\right) }\right) , \end{eqnarray*}
</div>
<p>where \(C{\gt}1\) is a constant. So, we have</p>
<div class="equation" id="***">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( D_{n}g\right) ^{\prime }-g^{\prime }\right\Vert \leq \tfrac {B}{\delta ^{2}}\tfrac {b_{n}^{3}}{n}\left( \left\Vert g^{\prime }\right\Vert +\left\Vert g^{\prime \prime \prime }\right\Vert \right) \label{***} \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>where \(B{\gt}1\) is a constant. This finally establishes the theorem. <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><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="Chl">1</a></dt>
  <dd><p><i class="sc">I. Chlodovsky</i>, <i class="it">Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M. S. Bernstein</i>, Compositio Math., <b class="bf">4</b> (1937), 380–393. </p>
</dd>
  <dt><a name="Hermann">2</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf01895859"> <i class="sc">T. Hermann</i>, <i class="it">Approximation of unbounded functions on unbounded interval</i>, Acta. Math. Hungar., <b class="bf">29</b> (1977), 393–398. <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="Butzer">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/1385-7258(80)90027-x"> <i class="sc">G. Bleimann, P.L. Butzer</i>, <i class="sc">L. Hahn</i>, <i class="it">A Bernstein-type operator approximating continuous functions on the semi-axis</i>, Indag. Math., <b class="bf">42</b> (1980), 255–262. <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="Durr">4</a></dt>
  <dd><p><i class="sc">J.L. Durrmeyer</i>, <i class="it">Une formule d’inversion de la transformée de Laplace: Applications à la théorie des moments</i>, Thèse de 3e Cycle, Faculté des Sciences de l’Université de Paris, 1967. </p>
</dd>
  <dt><a name="kt">5</a></dt>
  <dd><p><i class="sc">H. Karsli</i>, <i class="sc">P. Pych-Taberska,</i> <i class="it">On the rates of convergence of Chlodovsky–Durrmeyer operators and their Bézier variant</i>, Georgian Math. J., <b class="bf">16</b> (2009) no. 4, 693–704. </p>
</dd>
  <dt><a name="Kar">6</a></dt>
  <dd><p><i class="sc">H. Karsli</i>, <i class="it">Order of convergence of Chlodowsky type Durrmeyer operators for functions with derivatives of bounded variation</i>, Indian J. Pure Appl. Math., <b class="bf">38</b> (2007) no. 5, 353. </p>
</dd>
  <dt><a name="Lorentz">7</a></dt>
  <dd><p><i class="sc">G.G. Lorentz</i>, <i class="it">Bernstein polynomials</i>, University of Toronto Press, Toronto (1953). </p>
</dd>
  <dt><a name="BB">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1524/anly.2003.23.4.299"> <i class="sc">C. Bardaro, P.L. Butzer, R.L. Stens</i>, <i class="sc">G. Vinti</i>, <i class="it">Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals</i>, Analysis (Munich), <b class="bf">23</b> (2003) no. 4, 299–346. <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="Agratini">9</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.aml.2005.12.007"> <i class="sc">O. Agratini</i>, <i class="it">On the variation detracting property of a class of operators</i>, Appl. Math. Lett., <b class="bf">19</b> (2006), 1261–1264. <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="Kiv">10</a></dt>
  <dd><p><a href ="https://doi.org/10.3176/proc.2011.2.03"> <i class="sc">A. Kivinukk, T. Metsmagi</i>, <i class="it">Approximation in variation by the Meyer-König and Zeller operators</i>, Proc. Estonian Acad. Sci., <b class="bf">60</b> (2011) no. 2, 88–97. <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="Karsli">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00009-012-0186-4 "> <i class="sc">H. Karsli</i>, <i class="it">On convergence of Chlodovsky and Chlodovsky-Kantorovich polynomials in the variation seminorm</i>, Mediterr. J. Math., <b class="bf">10</b> (2013), 41–56. <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="Oz">12</a></dt>
  <dd><p><i class="sc">Ö. Öksüzer, H. Karsli, F. Tasdelen</i>, <i class="it">On convergence of Bernstein-Stancu polynomials in the variation seminorm</i>, Numer. Funct. Anal. Optim., <b class="bf">37</b> (2016) no. 4, 1–20. </p>
</dd>
  <dt><a name="Gul">13</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00009-015-0640-1"> <i class="sc">H.Gül  Ince  Ilarslan, G. Başcanbaz-Tunca</i>, <i class="it">Convergence in variation for Bernstein-type operators</i>, Mediterr. J. Math., <b class="bf">13</b> (2015) no. 5, pp. 2577–2592. <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="Abel">14</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10474-016-0642-x"> <i class="sc">U. Abel</i>, <i class="sc">O. Agratini</i>, <i class="it">On the variation detracting property of operators of Balázs and Szabados</i>, Acta Math. Hungarica, <b class="bf">150</b> (2016) no. 2, 383–395. <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="OKT">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00025-017-0653-0"> <i class="sc">H. Karsli, Ö. Öksüzer Yılık, F. Tasdelen</i>, <i class="it">Convergence of the Bernstein–Durrmeyer operators in variation seminorm</i>, Results Math., <b class="bf">72</b> (2017), 1257–1270. <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="BK">16</a></dt>
  <dd><p><i class="sc">P.L. Butzer</i>, <i class="sc">H. Karsli</i>, <i class="it">Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Száasz-Mirakyan operator</i>, Comment. Math., <b class="bf">49</b> (2009) no. 1, 33–57. </p>
</dd>
  <dt><a name="RM">17</a></dt>
  <dd><p><i class="sc">R.M. Trigub</i>, <i class="sc">E.S. Belinsky</i>, <i class="it">Fourier Analysis and Approximation of Functions</i>. Kluwer Academic Publishers, Dordrecht (2004). </p>
</dd>
</dl>


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

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

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