<!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>Better approximation by Stancu Beta operators in compact interval: Better approximation by Stancu Beta operators in compact interval</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>Better approximation by Stancu Beta operators in compact interval</h1>
<p class="authors">
<span class="author">Vijay Gupta\(^\ast \) Rani Yadav\(^\ast \)</span>
</p>
<p class="date">January 7, 2011</p>
</div>
<p>\(^\ast \)School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India, e-mail: <span class="ttfamily">{vijaygupta2001@hotmail.com, raniyadav23@gmail.com}</span>. </p>

<div class="abstract"><p> The present paper deals with the study of Stancu-Beta operators which preserve the constant as well as linear functions but not the quadratic ones. We apply the King’s approach to propose the modified form of these operators, so as they preserve the quadratic functions, which results in better approximation for the modified operators in the compact interval \((0,1)\) for these operators. </p>
<p><b class="bf">MSC.</b> </p>
<p><b class="bf">Keywords.</b> Positive linear operators, Korovkin-type approximation theorem, Stancu-Beta operators. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Many well-known approximating operators reproduce constant as well as linear functions for example the classical Bernstein polynomials, the Szász-Mirakjan operators, the Baskakov operators, and so on. In <span class="cite">
	[
	<a href="#UV" >1</a>
	]
</span> the authors have studied the rate of convergence for the well known Stancu Beta operators, which also reproduce constant and linear functions. But the operators do not preserve quadratic functions. In this case a natural question arises: can we modify these operators such that the quadratic functions are preserved? In this paper we mainly focus on this problem and find affirmative answers. Actually, the basic reason of this idea is to make convergence faster to the function being approximated. King <span class="cite">
	[
	<a href="#King" >3</a>
	]
</span> was the first, who considered the Bernstein polynomials and obtained the faster convergence by modifying the well known Bernstein polynomials. Here we study the convergence behavior of Stancu Beta operators. </p>
<p>D. D. Stancu <span class="cite">
	[
	<a href="#Stancu" >5</a>
	]
</span> introduced Beta operators \(L_n\) of second kind in order to approximate the Lebesgue integrable functions on the interval \((0,\infty )\) as </p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation} (L_nf)(x) = \tfrac {1}{B(nx,n+1)}\int _0^\infty \tfrac {t^{nx-1}}{(1+t)^{nx+n+1}}f(t){\rm d}t.\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> Then, Abel and Gupta in <span class="cite">
	[
	<a href="#UV" >1</a>
	]
</span>, obtained the rate of convergence by means of decomposition technique. The moments of the operators (1) are given in the following lemmas: </p>
<p><div class="lemma_thmwrapper " id="a0000000004">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">1</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#Stancu" >5</a>
	]
</span> For all \(e_i(x)=x^i, i\in \mathbb {N}\cup \left\{  0\right\} \), \(n\in \mathbb {N}\), \(x{\gt}0\) with \(n{\gt}i\), we have \((L_{n}e_{0})(x)=1\), \((L_{n}e_{1})(x)=e_{1}(x)\) and \((L_ne_2)(x)=x^2+\tfrac {x(1+x)}{n-1}.\) Also we have the recurrence relation </p>
<div class="displaymath" id="a0000000005">
  \[ (L_{n}e_{i+1})(x) =\tfrac {n x+i}{n-i}(L_{n}e_{i})(x), \mbox{ for all } n {\gt} i. \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>By the relationship (1) of the Beta operators of second kind, it is obvious that \((L_{n}e_{0})(x)=1\) and \((L_{n}e_{1})(x)=e_{1}(x)\) (see <span class="cite">
	[
	<a href="#UV" >1</a>
	]
</span>, Proposition 2). Next </p>
<div class="displaymath" id="a0000000007">
  \begin{align*}  (L_{n}e_{i+1})(x)& =\tfrac {1}{B(nx,n+1)}B(nx+i+1,n-i)=(L_{n}e_{i})(x)\cdot \tfrac {B(nx+i+1, n-i)}{B(nx+i, n-i+1)}\\ & =(L_{n}e_{i})(x)\cdot \tfrac {n x+i}{n-i}. \end{align*}
</div>
<p> Since \(B(\alpha , \beta )\) is only defined for \(\alpha {\gt}0\) and \(\beta {\gt}0\), it follows that the above recurrence is valid only for \(n-i{\gt}0\). The value of \((L_ne_2)(x)\) follows from the recurrence relation.<br />This completes the proof of Lemma 1. </p>
<p>By simple computation and using Lemma 1, we have the following result: <div class="lemma_thmwrapper " id="a0000000008">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For fixed \(x\in (0,\infty )\), if define the function \(\varphi _x\) by \(\varphi _x(t)=t-x.\) Then </p>
<ol class="enumerate">
  <li><p>\((L_n\varphi _x^0)(x)=1,\) </p>
</li>
  <li><p>\((L_n\varphi _x^1)(x)=0,\) </p>
</li>
  <li><p>\((L_n\varphi _x^2)(x)=\tfrac {x(1+x)}{n-1}.\) </p>
</li>
</ol>

  </div>
</div> </p>
<h1 id="a0000000009">2 Direct results</h1>
<p>In this section we compute the rates of convergence of the operators <br />\((L_nf)(x)\). We define by \(C_B(0,\infty ),\) the space of all bounded and continuous functions on \((0,\infty )\) endowed with the norm \(||f||=\sup _{x\in (0,\infty )}|f(t)|\). For \(f\in C_B(0,\infty )\) the first and second order modulus of continuity of \(f\) are denoted by \(\omega (f,\delta )\) and \(\omega _2(f,\delta )\) respectively and defined as </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \omega (f,\delta )=\sup _{x-\delta \leq t\leq x+\delta ,\, t\in (0,\infty )\text{ }}\left| f(t)-f(x)\right| , \end{equation*}
</div>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  \omega _2 (f,\sqrt{\delta })=\sup _{x-\delta \leq t\leq x+\delta ,\, t\in (0,\infty )\text{ }}\left| f(x+2h)-2f(x+h)+f(x)\right| . \end{equation*}
</div>
<p> The Peetre’s \(K\)-functional is defined as </p>
<div class="displaymath" id="a0000000012">
  \[ K_2(f,\delta )=\inf \{ ||f-g||+\delta ||g''|| : g\in W_\infty ^2\} , \delta {\gt}0, \]
</div>
<p> where \(W_{\infty }^2=\{ g\in C_B(0,\infty ):g',g''\in C_B(0,\infty )\} .\) Using <span class="cite">
	[
	<a href="#Lorentz" >2</a>
	]
</span>, there exists a positive constant \(C\) such that<br /></p>
<div class="displaymath" id="a0000000013">
  \[ K_2(f,\delta )\leq C\omega _2(f,\sqrt\delta ). \]
</div>
<p> <div class="theorem_thmwrapper " id="a0000000014">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For the Stancu-Beta operators we can write that, for every \(f\in C_B(0,\infty ), x{\gt}0\) and \(n{\gt}1,\) </p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert (L_nf)(x)-f(x)\right\vert \leq 2\omega (f,\alpha _x), \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \(\alpha _x=\sqrt{\tfrac {x(1+x)}{n-1}}.\) </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>For every \(\alpha {\gt}0\) and \(n\in N\), using linearity and monotonicity of \(L_n\) we easily get that </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  \left\vert (L_nf)(x)-f(x)\right\vert \leq \omega (f,\alpha )\left[1+\tfrac {1}{\alpha }\sqrt{(L_n\varphi _x^2)(x)}\right]. \end{equation*}
</div>
<p>Applying Lemma 2 and choosing \(\alpha =\alpha _x\), the proof is completed. </p>
<p><div class="theorem_thmwrapper " id="a0000000018">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C_B(0,\infty ),\) then for every \(x\in (0,\infty )\) and for \(C{\gt}0,\) we have </p>
<div class="displaymath" id="a0000000019">
  \[ |(L_nf)(x)-f(x)| \leq C\omega _2\left(f,\sqrt{\tfrac {x(1+x)}{n-1}}\right). \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(g\in W_\infty ^2.\) By Taylor’s expansion </p>
<div class="displaymath" id="a0000000021">
  \[ g(t)=g(x)+g'(x)(t-x)+\int _x^t (t-u)g''(u){\rm d}u, \]
</div>
<p> and Lemma 2, we have<br /></p>
<div class="displaymath" id="a0000000022">
  \[ (L_nf)(x)-g(x)=\left(L_n\int _x^t(t-u)g''(u){\rm d}u\right)(x), \]
</div>
<p> we know that </p>
<div class="displaymath" id="a0000000023">
  \[ \left|\int _x^t(t-u)g''(u){\rm d}u\right|\leq (t-u)^2||g''||, \]
</div>
<p> therefore </p>
<div class="displaymath" id="a0000000024">
  \[ |(L_nf)(x)-f(x)|\leq \left(L_n(t-u)^2(x)\right)||g''||=\tfrac {x(1+x)}{n-1}||g''||, \]
</div>
<p> by Lemma 1, we have </p>
<div class="displaymath" id="a0000000025">
  \[ |(L_nf)(x)| \leq \tfrac {1}{B(nx,n+1)}\int _0^\infty \tfrac {t^{nx-1}}{(1+t)^{nx+n+1}}|f(t)|{\rm d}t\leq ||f||. \]
</div>
<p> Hence </p>
<div class="displaymath" id="a0000000026">
  \begin{align*}  |(L_nf)(x)-f(x)|& \leq \left|\left(L_n(f-g)\right)(x)-(f-g)(x)\right|+|(L_nf)(x)-f(x)|\\ & \leq 2||f-g||+\tfrac {x(1+x)}{n-1}||g”||, \end{align*}
</div>
<p> taking the infimum on the right side over all \(g\in W_\infty ^2\) and using \(K\)-functional, the required result is obtained. </p>
<h1 id="a0000000027">3 Construction of operators</h1>
<p> Let \({r_n(x)}\) be sequence of real valued continuous functions defined on \((0,\infty )\) with \(0{\lt}r_n(x){\lt}\infty )\) and defined by </p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  r_{n}(x)=\tfrac {-1+\sqrt{1+4n(n-1)x^2}}{2n}. \end{equation*}
</div>
<p> Then we can define the modified form of operators (1) as </p>
<div class="equation" id="a0000000029">
<p>
  <div class="equation_content">
    \begin{equation}  (L_n^*f)(x) = \tfrac {1}{B(nr_n(x),n+1)}\int _0^\infty \tfrac {t^{nr_n(x)-1}}{(1+t)^{nr_n(x)+n+1}}f(t){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>For these modified operators, we have the following lemmas: <div class="lemma_thmwrapper " id="a0000000030">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For each \(x{\gt}0\), we have </p>
<ol class="enumerate">
  <li><p>\((L_n^*1)(x)=1,\) </p>
</li>
  <li><p>\((L_n^*t)(x)=\tfrac {-1+\sqrt{1+4n(n-1)x^2}}{2n},\) </p>
</li>
  <li><p>\((L_n^*t^2)(x)=x^2.\) </p>
</li>
</ol>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000031">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For fixed \(x\in (0,\infty )\), define the function \(\varphi _x\) by \(\varphi _x(t)=t-x.\) The central moments for the operators \(L_n^* \) are given by </p>
<ol class="enumerate">
  <li><p>\((L_n^*\varphi _x^0)(x)=1,\) </p>
</li>
  <li><p>\((L_n^*\varphi _x^1)(x)=-x+\tfrac {-1+\sqrt{1+4n(n-1)x^2}}{2n},\) </p>
</li>
  <li><p>\((L_n^*\varphi _x^2)(x)=2x\left[x+\tfrac {1}{2n}-\tfrac {\sqrt{1+4n(n-1)x^2}}{2n}\right].\) </p>
</li>
</ol>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="thm31">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For every \(f\in C_B(0,\infty )\), \(x{\gt}0\) and \(n{\gt}1\), we have </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  \left\vert (L_n^*f)(x)-f(x)\right\vert \leq 2\omega (f,\delta _x), \end{equation*}
</div>
<p>where </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  \delta _x=2x\left[x+\tfrac {1}{2n}-\tfrac {1}{2n}\sqrt{1+4n(n-1)x^2}\right]. \end{equation*}
</div>

  </div>
</div> The proof of this can be carried out on the same lines as of Theorem 3. </p>
<p><div class="remark_thmwrapper " id="a0000000034">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">8</span>
  </div>
  <div class="remark_thmcontent">
  <p>Now considering the above remark the similar claim is valid for the operators \(L_n^*\) on the interval \((0,1) .\) In order to get a better estimation we must show that \(\delta _x{\lt} \alpha _x\) for appropriate \(x\)’s. Indeed, for \(0{\lt}x{\lt}1\), we have \(x^2{\lt}1\). Also since </p>
<div class="displaymath" id="a0000000035">
  \[ x^2\left[(2n-1)^2-4n(n-1)\right]{\lt} 1, \]
</div>
<p> or </p>
<div class="displaymath" id="a0000000036">
  \[ 1+4n(n-1)x^2{\gt}(2n-1)^2x^2, \]
</div>
<p> which gives </p>
<div class="displaymath" id="a0000000037">
  \[ \sqrt{1+4n(n-1)x^2}{\gt}(2n-1)x, \]
</div>
<p> then we obtain </p>
<div class="displaymath" id="a0000000038">
  \[ -\tfrac {1}{2n}+\tfrac {1}{2n}\sqrt{1+4n(n-1)x^2}{\gt}-\tfrac {1}{2n}+\tfrac {2n-1}{2n}x, \]
</div>
<p> thus we have \(x-r_n(x){\lt}\tfrac {1+x}{2n},\)<br />i.e. </p>
<div class="displaymath" id="a0000000039">
  \[ 2x(x-r_n(x)){\lt}\tfrac {x(1+x)}{n}{\lt}\tfrac {x(1+x)}{n-1}, \]
</div>
<p> for \(x\in (0,1)\) and \(n{\gt}1.\) This guarantees that \(\delta _x{\lt}\alpha _x\) for \(x\in (0,1)\) and \(n{\gt}1\), which corrects our claim. </p>
<p>A function \(f\in C_B(0,\infty )\) belongs to \({\rm Lip}_M(\alpha ),\) if the following inequality holds, </p>
<div class="equation" id="a0000000040">
<p>
  <div class="equation_content">
    \begin{equation}  \left|f(y)-f(x)\right|\leq M|y-x|^\alpha (x,y\in (0,\infty )). \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000041">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For every \(f\in {\rm Lip}_M(\alpha ), x{\gt}0\) and \(n{\gt}1,\) we have </p>
<div class="displaymath" id="a0000000042">
  \[ |(L_n^*f)(x)-f(x)|\leq M\left\{ 2x\left(x+\tfrac {1-\sqrt{1+4n(n-1)x^2}}{2n}\right)\right\} ^{\alpha /2}. \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000043">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Since \(f\in {\rm Lip}_M(\alpha )\) and \(x{\gt}0\), using (4) and then applying the Holder’s inequality with \(p=\alpha ,q=\tfrac {2}{2-\alpha }\), we get </p>
<div class="displaymath" id="a0000000044">
  \begin{align*}  |(L_n^*f)(x)-f(x)|& \leq \left(L_n^*|f(y)-f(x)|\right)(x)\\ & \leq M(L_n^*|y-x|^\alpha )(x)\leq M\{ (L_n^*\varphi _x^2)(x)\} ^{\alpha /2}, \end{align*}
</div>
<p> using Lemma 6, we get the required result. </p>
<p>Consider the class </p>
<div class="displaymath" id="a0000000045">
  \[ C_\gamma (0,\infty )=\{ f\in C(0,\infty ):|f(x)|\leq M(1+x)^\gamma \  {\rm for\  some}\  M{\gt}0,\gamma {\gt}0\} , \]
</div>
<p> with the norm </p>
<div class="displaymath" id="a0000000046">
  \[ ||f||_\gamma =\sup _{x\in (0,\infty )}\tfrac {|f(x)|}{(1+x)^\gamma }. \]
</div>
<p> Let \(C_\gamma ^{(m)}(0,\infty ), m=0,1,2,...,\) where \(f\in C_\gamma (0,\infty ).\) Following <span class="cite">
	[
	<a href="#KP" >4</a>
	]
</span>, we consider the m-th order generalization of the positive linear operators \(L_n^*\) as </p>
<div class="equation" id="a0000000047">
<p>
  <div class="equation_content">
    \begin{equation}  (L_{n,m}^*f)(x)=\tfrac {1}{B(nr_n(x),n+1)}\sum _{i=0}^m \int _0^\infty \tfrac {t^{nr_n(x)-1}}{(1+t)^{nr_n(x)+n+1}}f^{(i)}(t)\tfrac {(t-x)^i}{i!}{\rm d}t. \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> Here \((L_{n,0}^*f)(x)=(L_n^*f)(x),\) and </p>
<div class="equation" id="a0000000048">
<p>
  <div class="equation_content">
    \begin{equation}  (L_{n,m}^*f)(x)=\sum _{i=0}^m \int _0^\infty V_n(x,t)f^{(i)}(t)\tfrac {(t-x)^i}{i!}{\rm d}t, \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000049">
  \[ V_n(x,t)=\tfrac {1}{B(nr_n(x),n+1)}\tfrac {t^{nr_n(x)-1}}{(1+t)^{nr_n(x)+n+1}}. \]
</div>
<p> <div class="theorem_thmwrapper " id="a0000000050">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For all \(f\in C_\gamma ^{(m)}(0,\infty ),\gamma {\gt}0,\) such that \(f^{(m)}\in {\rm Lip}_M(\alpha ),\) and for every \(x{\gt}0\) we have </p>
<div class="displaymath" id="a0000000051">
  \[ |(L_{n,m}^*f)(x)-f(x)|\leq \tfrac {M}{(m-1)!}\tfrac {\alpha }{\alpha +m}B(\alpha ,m)|(L_n^*|t-x|^{m+\alpha })(x)|, \]
</div>
<p> where \(m=1,2,...\) and \(B(\alpha ,m)\) is the beta function. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000052">
  <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 (6) and Lemma 5 we have </p>
<div class="equation" id="a0000000053">
<p>
  <div class="equation_content">
    \begin{equation}  f(x)-(L_{n,m}^*f)(x)=\int _0^\infty V_n(x,t)\{ f(x)-\sum _{i=0}^mf^{(i)}(t)\tfrac {(t-x)^i}{i!}\} {\rm d}t. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> By Taylor’s formula </p>
<div class="displaymath" id="a0000000054">
  \begin{align*} & f(x)-\sum _{i=0}^mf^{(i)}(t)\tfrac {(x-t)^i}{i!}=\\ & =\tfrac {(x-t)^m}{(m-1)!}\int _0^1(1-s)^{m-1}\left\{ f^{(m)}(t+s(x-t))-f^{(m)}(t)\right\} {\rm d}s. \end{align*}
</div>
<p>Since \(f^{(m)}\in {\rm Lip}_m(\alpha ),\) </p>
<div class="equation" id="a0000000055">
<p>
  <div class="equation_content">
    \begin{equation}  |f^{(m)}(t+s(x-t))-f^{(m)}(t)|\leq Ms^{\alpha }|t-x|^{\alpha }. \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> From above equation and by Beta integral, we get </p>
<div class="equation" id="a0000000056">
<p>
  <div class="equation_content">
    \begin{equation}  |f(x)-\sum _{i=0}^mf^{(m)}(t)\tfrac {(x-t)^i}{i!}|\leq \tfrac {M}{(m-1)!}\tfrac {\alpha }{\alpha +m}B(\alpha ,m)|t-x|^{\alpha +m}. \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> Hence the proof. Finally, for the uniform convergence of the operators \(L_{n,m}^*\) we obtain the following result: </p>
<p><div class="theorem_thmwrapper " id="a0000000057">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">11</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For every \(f\in C_\gamma ^{(m)}(0,\infty ), \gamma {\gt}0, m=1,2,...,\) such that \(f^{(m)}\in {\rm Lip}_M(\alpha ),\) we have</p>
<div class="displaymath" id="a0000000058">
  \[ \displaystyle \lim _{n\rightarrow \infty }(L_{n,m}^*f)(x)=f(x) \]
</div>
<p> uniformly with respect to \(x\in (0,\infty ).\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000059">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Define the function \(g\) by \(g(t)=|t-x|^{m+\alpha }.\) Then by Theorem 4, we have </p>
<div class="displaymath" id="a0000000060">
  \[ \displaystyle \lim _{n\rightarrow \infty }(L_{n,m}^*f)(x)=f(x)=0 \]
</div>
<p> uniformly with respect to \(x\in (0,\infty ).\) Thus the proof is followed by previous theorem. </p>
<h1 id="a0000000061">4 concluding remarks</h1>
<p> By applying this approach the modified Stancu Beta operators \((L_n^*f)(x)\) preserve the test function \(e_2(x)\) but the modified operators do not preserve the test function \(e_{1}(x)\). Also we can have better approximation on the compact interval \((0,1)\), while the modified operators are defined on the interval \((0,\infty ).\) </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000062">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors are thankful to the referee(s) for valuable suggestions, leading to the overall improvements in the paper. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="UV">1</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art1"> <i class="sc">U. Abel</i> and <i class="sc">V. Gupta</i>, <i class="it">Rate of convergence of Stancu Beta operators for functions of bounded variation</i>, Rev. Anal. Number. Théor. Approx., <b class="bf">33</b> (1), pp.&#160;3–9, 2004. <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="Lorentz">2</a></dt>
  <dd><p><i class="sc">R. A. Devore</i> and <i class="sc">G. G. Lorentz</i>, <i class="it">Costructive approximation</i>, Springer, Berlin, 1993. </p>
</dd>
  <dt><a name="King">3</a></dt>
  <dd><p><i class="sc">J. P. King</i>, <i class="it">Positive linear operators which preserve \(x^2\)</i>, Acta Math. Hungar., <b class="bf">99</b>(3), pp.&#160;203–208, 2003. </p>
</dd>
  <dt><a name="KP">4</a></dt>
  <dd><p><i class="sc">G. Kirov,</i> and <i class="sc">L. Popova</i>, <i class="it">A generalization of the linear positive operators</i>, Mathematica Balkanica, N.S., <b class="bf">7</b>(2), pp.&#160;149–162, 1993. </p>
</dd>
  <dt><a name="Stancu">5</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art26"> <i class="sc">D. D. Stancu</i>, On the beta approximating operators of second kind, Rev. Anal. Num. Théor. Approx., <b class="bf">2</b>(4), pp.&#160;231–239, 1995. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </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>