<!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 the composite Bernstein type quadrature formula: On the composite Bernstein type quadrature formula</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 the composite Bernstein type quadrature formula</h1>
<p class="authors">
<span class="author">Dan Bărbosu\(^\ast \) Dan Miclăuş\(^\ast \)</span>
</p>
<p class="date">September 23, 2009.</p>
</div>
<p>\(^\ast \)Department of Mathematics and Computer Science, Faculty of Sciences, North University of Baia Mare, Victoriei 76, 430122 Baia Mare, Romania, e-mail: <br /><span class="tt">{barbosudan,danmiclausrz}@yahoo.com</span>. </p>

<div class="abstract"><p> Considering a given function \(f\in C[0,1]\), the interval \([0,1]\) is divided in \(m\) equally spaced subintervals \(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\). On each of such type of interval the Bernstein approximation formula is applied and a corresponding Bernstein type quadrature formula is obtained. Making the sum of mentioned quadrature formulas, the composite Bernstein type quadrature formula is obtained. </p>
<p><b class="bf">MSC.</b> 65D32, 41A10 </p>
<p><b class="bf">Keywords.</b> Bernstein operator, Bernstein approximation formula, Bernstein quadrature formula, divided differences, remainder term. </p>
</div>
<h1 id="a0000000002">1 Preliminaries</h1>
<p>Let \(\mathbb {N}\) be the set of positive integers and \(\mathbb {N}_{0}=\mathbb {N}\cup \{ 0\} \). </p>
<p>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 _{j=0}^{n}p_{n,j}(x)f\left(\tfrac {j}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p> where \(p_{n,j}\) 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,j}(x)=\tbinom {n}{j}x^{j}(1-x)^{n-j}, \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p> for any \(x\in [0,1]\), any \(j\in \{ 0,1,\ldots ,n\} \) and any \(n\in \mathbb {N}\), are called Bernstein operators and were first introduced in [3]. The approximation properties of the Bernstein operator were intensively studied in [1], [4], [5]. </p>
<p>For any \(f\in C[0,1]\), any \(x\in [0,1]\) and any \(n\in \mathbb {N}\), the following equality </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.3} f(x)=(B_{n}f)(x)+(R_{n}f)(x) \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> is called the Bernstein approximation formula, where \(R_n\) is the remainder operator associated to the Bernstein operator \(B_n\), i.e. \(R_{n}f\) is the remainder term of the approximation formula <a href="#1.3" class="eqref">1.3</a>. Regarding the remainder term of <a href="#1.3" class="eqref">1.3</a>, Tiberiu Popoviciu [4] established the following: </p>
<p><div class="theorem_thmwrapper " id="a0000000003">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any \(f\in C[0,1]\) there exist the distinct points \(\xi _{1},\xi _{2},\xi _{3}\in [0,1]\) such that, for any \(x\in [0,1]\), the remainder term of <a href="#1.3" class="eqref">1.3</a> can be represented under the form </p>
<div class="equation" id="1.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.4} (R_{n}f)(x)=-\tfrac {x(1-x)}{n}[\xi _{1},\xi _{2},\xi _{3};f]. \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>

  </div>
</div> In <a href="#1.4" class="eqref">1.4</a> the brackets denote the divided difference of function \(f\) with respect the distinct knots \(\xi _{1},\xi _{2},\xi _{3}\). It is well known the following estimation of the remainder term of <a href="#1.3" class="eqref">1.3</a>, (see [7]). </p>
<p><div class="theorem_thmwrapper " id="a0000000004">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose that \(f\in C^{2}[0,1]\). The following inequality </p>
<div class="equation" id="1.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.5} |(R_{n}f)(x)|\leq \tfrac {x(1-x)}{2n}M_{2}[f] \end{equation}
  </div>
  <span class="equation_label">1.5</span>
</p>
</div>
<p> holds, where </p>
<div class="equation" id="1.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.6} M_{2}[f]=\max \limits _{x\in [0,1]}|f^{\prime \prime }(x)|. \end{equation}
  </div>
  <span class="equation_label">1.6</span>
</p>
</div>

  </div>
</div> </p>
<p>The inequality <a href="#1.5" class="eqref">1.5</a> follows directly from <a href="#1.4" class="eqref">1.4</a>, applying the mean value theorem for divided differences and it is attributed to D. D. Stancu. </p>
<p>In the following we suppose that \(f\in C^{2} [0,1]\). Starting with the Bernstein approximation formula <a href="#1.3" class="eqref">1.3</a>, in [7] the following Bernstein quadrature formula </p>
<div class="equation" id="1.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.7} \int _{0}^{1}f(x){\rm d}x=\sum _{j=0}^{n}A_{j}f\left(\tfrac {j}{n}\right)+R_{n}[f] \end{equation}
  </div>
  <span class="equation_label">1.7</span>
</p>
</div>
<p> is obtained, where </p>
<div class="equation" id="1.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.8} A_{j}=\tfrac {1}{n+1}, \   (\forall ) \   j=\overline{0,n} \end{equation}
  </div>
  <span class="equation_label">1.8</span>
</p>
</div>
<p> and </p>
<div class="equation" id="1.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.9} |R_{n}[f]|\leq \tfrac {1}{12n}M_{2}[f]. \end{equation}
  </div>
  <span class="equation_label">1.9</span>
</p>
</div>
<p> The focus of the present paper is to construct the composite Bernstein type quadrature formula. For this aim, the interval \([0,1]\) will be divided in \(m\) equally spaced subintervals \(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\). On each of such type of interval, the Bernstein quadrature formula <a href="#1.7" class="eqref">1.7</a> will be applied. Next, adding the mentioned quadrature formulas, the desired Bernstein type quadrature formula on \([0,1]\) will be obtained. </p>
<h1 id="a0000000005">2 Main results</h1>
<p> We start with two auxiliary results. <div class="lemma_thmwrapper " id="a0000000006">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Suppose that \(a,b\in \mathbb {R}\), \(a{\lt}b\) and \(f\in C[a,b]\). Then, the Bernstein polynomial associated to the function \(f\) is defined by </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} (B_{n}f)(x)=\tfrac {1}{(b-a)^n}\sum _{j=0}^{n}\tbinom {n}{j}(x-a)^{j}(b-x)^{n-j}f\left(a+j\tfrac {b-a}{n}\right). \end{equation}
  </div>
  <span class="equation_label">2.10</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000007">
  <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 is easy to observe that the correspondence \(t\to \tfrac {x-a}{b-a}\) transform the interval \([a,b]\) in the interval \([0,1]\). Taking <a href="#1.1" class="eqref">1.1</a>, <a href="#1.2" class="eqref">1.2</a> and the above remark into account, yields <a href="#2.1" class="eqref">2.10</a>. </p>
<p><div class="lemma_thmwrapper " id="a0000000008">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Suppose that \(a,b\in \mathbb {R}\), \(a{\lt}b\) and \(f\in C^{2}[a,b]\). Then, the remainder term of the Bernstein approximation formula on \([a,b]\) verifies the inequality </p>
<div class="equation" id="2.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.2} |(R_{n}f)(x)|\leq \tfrac {(x-a)(b-x)}{2n(b-a)^{2}}M_{2}[f], \end{equation}
  </div>
  <span class="equation_label">2.11</span>
</p>
</div>
<p> where \(M_{2}[f]\) is defined at <a href="#1.6" class="eqref">1.6</a>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>One applies in a way similar to the case of relation <a href="#1.5" class="eqref">1.5</a>, taking the transformation \(t\to \tfrac {x-a}{b-a}\) into account. In what follows, let us to consider the interval \([0,1]\) divided in the equally spaced subintervals \(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\). In each interval \(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\), one considers the distinct knots \(x_{i}=\tfrac {kn-n+i}{mn}\), \(i=\overline{0,n}\). Applying Lemma 2.1., yields the following Bernstein type polynomial </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} (B_{n,k}f)(x)=m^{n}\sum _{i=0}^{n}\tbinom {n}{i}\left(x-\tfrac {k-1}{m}\right)^{i}\left(\tfrac {k}{m}-x\right)^{n-i}f\left(\tfrac {kn-n+i}{mn}\right). \end{equation}
  </div>
  <span class="equation_label">2.12</span>
</p>
</div>
<p> The corresponding Bernstein type approximation formula on the interval <br />\(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\), becomes </p>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} f(x)=(B_{n,k}f)(x)+(R_{n,k}f)(x). \end{equation}
  </div>
  <span class="equation_label">2.13</span>
</p>
</div>
<p> If \(f\in C^{2}[0,1]\), the remainder term of <a href="#2.4" class="eqref">2.13</a> verifies the inequality </p>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.5} |(R_{n,k}f)(x)|\leq \tfrac {\left(x-\tfrac {k-1}{m}\right)\left(\tfrac {k}{m}-x\right)}{2n}m^{2}M_{2}[f]. \end{equation}
  </div>
  <span class="equation_label">2.14</span>
</p>
</div>
<p><div class="theorem_thmwrapper " id="a0000000010">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f\in C^{2}[0,1]\), the following Bernstein type quadrature formula </p>
<div class="equation" id="2.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.6} \int _{\tfrac {k-1}{m}}^{\tfrac {k}{m}}f(x){\rm d}x=\sum _{i=0}^{n}A_{i,k}f\left(\tfrac {kn-n+i}{mn}\right)+R_{k}[f] \end{equation}
  </div>
  <span class="equation_label">2.15</span>
</p>
</div>
<p> holds, for any \(k=\overline{1,m}\), where </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} A_{i,k}=\tfrac {1}{m(n+1)}, \   (\forall ) \   i=\overline{0,n} \end{equation}
  </div>
  <span class="equation_label">2.16</span>
</p>
</div>
<p> and </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} |R_{k}[f]|\leq \tfrac {1}{12mn}M_{2}[f]. \end{equation}
  </div>
  <span class="equation_label">2.17</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Integrating <a href="#2.4" class="eqref">2.13</a> on \(\left[\tfrac {k-1}{m},\tfrac {k}{m}\right]\), \(k=\overline{1,m}\), and taking <a href="#2.3" class="eqref">2.12</a> into account, yields </p>
<div class="displaymath" id="a0000000012">
  \begin{align*}  A_{i,k} & =m^{n}\tbinom {n}{i}\int _{\tfrac {k-1}{m}}^{\tfrac {k}{m}}\left(x-\tfrac {k-1}{m}\right)^{i}\left(\tfrac {k}{m}-x\right)^{n-i}{\rm d}x\\ &  = m^{n}\tbinom {n}{i}\int _{0}^{1}\left(\tfrac {t}{m}\right)^i\left[\tfrac {1}{m}(1-t)\right]^{n-i}\tfrac {1}{m}{\rm d}t\\ & =\tfrac {1}{m}\tbinom {n}{i}\int _{0}^{1}t^{i}(1-t)^{n-i}{\rm d}t. \end{align*}
</div>
<p> The last integral is the Euler function of first kind \(B(i+1,n-i+1)\). Using the well known properties of Euler function of first kind, it follows </p>
<div class="displaymath" id="a0000000013">
  \begin{align*}  A_{i,k}=\tfrac {1}{m}\tbinom {n}{i}B(i+1,n-i+1)=\tfrac {1}{m}\tfrac {n!}{i!(n-i)!}\tfrac {i!(n-i)!}{(n+1)!}=\tfrac {1}{m(n+1)}. \end{align*}
</div>
<p> For the remainder term, taking <a href="#2.5" class="eqref">2.14</a> into account, we get </p>
<div class="equation" id="2.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.9} |R_{k}[f]|\leq M_{2}[f]\tfrac {m^{2}}{2n}\int _{\tfrac {k-1}{m}}^{\tfrac {k}{m}}\left(x-\tfrac {k-1}{m}\right)\left(\tfrac {k}{m}-x\right){\rm d}x. \end{equation}
  </div>
  <span class="equation_label">2.18</span>
</p>
</div>
<p> Because </p>
<div class="displaymath" id="a0000000014">
  \[  \int _{\tfrac {k-1}{m}}^{\tfrac {k}{m}}\left(x-\tfrac {k-1}{m}\right)\left(\tfrac {k}{m}-x\right){\rm d}x=\tfrac {1}{6m^{3}}  \]
</div>
<p> and from <a href="#2.9" class="eqref">2.18</a> one arrives to the desired inequality <a href="#2.8" class="eqref">2.17</a>. </p>
<p><div class="theorem_thmwrapper " id="a0000000015">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any \(f\in C^{2}[0,1]\), the following composite Bernstein type quadrature formula </p>
<div class="equation" id="2.10">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.10} \int _{0}^{1}f(x){\rm d}x=\tfrac {1}{m(n+1)}\sum _{k=1}^{m}\sum _{i=0}^{n}f\left(\tfrac {kn-n+i}{mn}\right)+R_{n}[f] \end{equation}
  </div>
  <span class="equation_label">2.19</span>
</p>
</div>
<p> holds, where </p>
<div class="equation" id="2.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.11} |R_{n}[f]|\leq \tfrac {1}{2n}M_{2}[f]. \end{equation}
  </div>
  <span class="equation_label">2.20</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Adding the Bernstein type quadrature formulas <a href="#2.6" class="eqref">2.15</a> for \(k=\overline{1,m}\), we get the following composite Bernstein type quadrature formula </p>
<div class="equation" id="2.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.12} \int _{0}^{1}f(x){\rm d}x=\tfrac {1}{m(n+1)}\sum \limits _{k=1}^{m}\sum _{i=0}^{n}f\left(\tfrac {kn-n+i}{mn}\right)+\sum _{k=1}^{m}R_{k}[f]. \end{equation}
  </div>
  <span class="equation_label">2.21</span>
</p>
</div>
<p> Denoting \(R_{n}[f]=\sum \limits _{k=1}^{m}R_{k}[f]\) and taking <a href="#2.8" class="eqref">2.17</a> into account, yields </p>
<div class="equation" id="2.13">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.13} |R_{n}[f]|\leq \sum _{k=1}^{m}|R_{k}[f]|\leq \tfrac {1}{12mn}M_{2}[f]\cdot m=\tfrac {1}{12n}M_{2}[f]. \end{equation}
  </div>
  <span class="equation_label">2.22</span>
</p>
</div>

<p><div class="remark_thmwrapper " id="a0000000017">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.5</span>
  </div>
  <div class="remark_thmcontent">
  <p>It is easy to see that we get the same result for the remainder term of the composite Bernstein type quadrature formula as the result obtained by D. D. Stancu in [7], for the Bernstein quadrature formula.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="a0000000018">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">2.6</span>
  </div>
  <div class="corollary_thmcontent">
  <p>For any \(f\in C^{2}[0,1]\) and any \(m\in \mathbb {N}\), the following equality </p>
<div class="equation" id="2.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.14} \lim \limits _{n\to \infty }\tfrac {1}{m(n+1)}\sum _{k=1}^{m}\sum _{i=0}^{n}f\left(\tfrac {kn-n+i}{mn}\right)=\int _{0}^{1}f(x){\rm d}x \end{equation}
  </div>
  <span class="equation_label">2.23</span>
</p>
</div>
<p> holds. </p>

  </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> From <a href="#2.11" class="eqref">2.20</a> follows \(\lim \limits _{n\to \infty }R_{n}[f]=0\) and then, taking <a href="#2.10" class="eqref">2.19</a> into account one arrives to <a href="#2.14" class="eqref">2.23</a>. <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> </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000021">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors thank to professor Heiner Gonska for his remarks and suggestions regarding the first variant of the paper. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><i class="sc">Agratini, O.</i>, <em>Approximation by linear operators</em>, Presa Universitară Clujeană, 2000, (in Romanian). </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">Bărbosu, D.</i>, <em>A Schurer-Stancu type quadrature formula</em>, Carpathian J. Math., <b class="bf">23</b>, nos. 1–2, pp.&#160;27–31, 2007. </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><i class="sc">Bernstein, S.N.</i>, <em>Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités</em>, Commun. Soc. Math. Kharkow, <b class="bf">13</b>, no. 2, pp. 1–2, 1912–1913. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><i class="sc">Popoviciu, T.</i>, <em>Sur le rest dans certains formules lineaires d’approximation de l’analyse</em>, Mathematica <b class="bf">I</b>, <b class="bf">24</b>, pp.&#160;95–142, 1959. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">Stancu, D. D.</i>, <em>Quadrature formulas constructed by using certain linear positive operators</em>, Numerical Integration (Proc. Conf. Math. Res. Inst. Oberwolffach) (Basel) (G. Hammerlin, ed.), Birkhäuser, pp.&#160;241–251, 1982. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/1999-vol28-no1-art8"> <i class="sc">Stancu, D. D.</i> and <i class="sc">Vernescu, A.</i>, <em>On some remarkable positive polynomial operators of approximation</em>, Rev. Anal. Numér. Théor. Approx., <b class="bf">28</b>, pp.&#160;85–95, 1999. <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="7">7</a></dt>
  <dd><p><i class="sc">Stancu, D. D., Coman, Gh.</i> and <i class="sc">Blaga, P.</i>, <em>Numerical Analysis and Approximation Theory</em>, <b class="bf">II</b>, Presa Universitară Clujeană, 2002 (in Romanian). </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>