<!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>Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render: Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render</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>Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render</h1>
<p class="authors">
<span class="author">Ana-Maria Acu\(^1\), Heiner Gonska\(^2\) Margareta Heilmann\(^3\)</span>
</p>
<p class="date">received: August 13, 2021; accepted: September 16, 2021; published online: November 8, 2021.</p>
</div>
<div class="abstract"><p> The Bernstein-type operator of Aldaz, Kounchev and Render (2009) is discussed. New direct results in terms of the classical second order modulus as well as in a modification following Marsden and Schoenberg are given. </p>
<p><b class="bf">MSC.</b> 41A25, 41A36. </p>
<p><b class="bf">Keywords.</b> Bernstein-type operator; King operator; second order modulus of continuity; Marsden-Schoenberg; modulus of order \(j\). </p>
</div>
<p>This brief note is dedicated to our dear friend and long term collaborator <b class="bf">Ioan Raşa</b> on the occasion of his 70th birthday. Over the years he did a lot, most fruitful work on quite a number of Bernstein-type and related operators, among others. Here we deal with a then surprising, 12-year old object of this type offering a lot of challenges.  </p>
<p>\(^1\)Lucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No. 5-7, RO-550012 Sibiu, Romania, e-mail: <span class="tt">anamaria.acu@ulbsibiu.ro</span>. </p>
<p>\(^2\)University of Duisburg-Essen, Faculty of Mathematics, Bismarckstr. 90, D-47057 Duisburg, Germany, e-mail: <span class="tt">heiner.gonska@uni-due.de</span>. </p>
<p>\(^3\)University of Wuppertal, School of Mathematics and Natural Sciences, GauSSstraSSe 20, D-42119 Wuppertal, Germany, e-mail: <span class="tt">heilmann@math.uni-wuppertal.de</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Starting from the classical Bernstein operators \(B_n\) defined for \(f\in C[0,1]\) as </p>
<div class="displaymath" id="a0000000003">
  \[  B_n(f;x)=\displaystyle \sum _{k=0}^nf\left(\tfrac {k}{n}\right)p_{n,k}(x),\, \,  \text{ where} \]
</div>
<div class="displaymath" id="a0000000004">
  \[  p_{n,k}(x)=\textstyle {n\choose k}x^k(1-x)^{n-k},  \]
</div>
<p> during recent years many, probably much too many, modifications have been considered. One of the present hypes follows a 2003 paper written by J.P. King <span class="cite">
	[
	<a href="#1" >16</a>
	]
</span> in order to obtain linear positive operators which preserve two functions different from the classical test functions \(e_0\) and \(e_1\). Here we used the convention \(e_j(x)=x^j,\, \,  j=0,1,\dots \) </p>
<p>King modified the classical Bernstein operators as follows: </p>
<div class="equation" id="e1">
<p>
  <div class="equation_content">
    \begin{equation} \label{e1} f\to \left(B_nf\right)\circ r_n,\, \,  f\in C[0,1], \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000005">
  \[ r_n(x)=\left\{ \begin{array}{ll} x^2,&  n=1,\\ \tfrac {1}{2(n-1)}\left(-1+\sqrt{1+4n(n-1)x^2}\right), &  n=2,3,\dots \\ \end{array}\right.  \]
</div>
<p>These operators preserve the functions \(e_i(x)=x^i\) for \(i=0,2\). A slight extension was considered by Cárdenas <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#CGM2006" >6</a>
	]
</span> where a sequence of operators \(B_{n,\alpha }\) that preserve \(e_0\) and \(e_2+\alpha e_1\), \(\alpha \in [0,+\infty )\) was introduced. It is clear that the operator of King does not produce polynomials in the general case. Therefore, in their 2005 remarks on the article of King, Gonska and Piţul (see <span class="cite">
	[
	<a href="#GoPi2005" >15</a>
	]
</span>) asked the question if there exist linear and positive <b class="bf">polynomial</b> operators that reproduce \(e_2\). </p>
<p>Using a continuous strictly increasing function \(\tau \) defined on \([0,1]\) with \(\tau (0)=0\) and \(\tau (1)=1\), \(\tau '(x){\gt}0, x\in [0,1]\), Cárdenas-Morales <i class="it">et al.</i> (see <span class="cite">
	[
	<a href="#CGM2006" >6</a>
	]
</span>) introduced a modification of the Bernstein operator as follows: </p>
<div class="equation" id="e2">
<p>
  <div class="equation_content">
    \begin{equation} \label{e2} B_n^{\tau }(f;x):=\displaystyle \sum _{k=0}^n\textstyle {n\choose k}\tau (x)^k(1-\tau (x))^{n-k}(f\circ \tau ^{-1})\left(\tfrac kn\right), \, \,  f\in C[0,1],\, \,  x\in [0,1]. \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>Note that \(\tau '(0) {\gt} 0\) is essential for their results. A predecessor can be found in Cottin <i class="it">et al.</i> <span class="cite">
	[
	<a href="#CGGKZ1999" >9</a>
	]
</span>, but \(\tau (x)= \sqrt{x}\) there. <br /></p>
<p>These operators preserve the functions \(e_0\) and \(\tau \). So, for \(\tau (x)=x^j\), one has </p>
<div class="displaymath" id="a0000000006">
  \[  B_n^{\tau }(e_0;x)=e_0 \textrm{ and } B_n^{\tau }(e_j;x)=e_j. \]
</div>
<p> In this case \(B_n^{\tau }\) produces polynomials of degree \(jn.\) If the function \(\tau \) is not a polynomial, then the modified operator (<a href="#e2">2</a>) is not a polynomial operator. </p>

<p>Fix \(j{\gt}1\), \(j\in {\mathbb N}\). Let \(\Pi _n\) be the space of polynomials over \([0,1]\) of degree less than or equal to \(n\). For every \(n\geq j\), Aldaz, Kounchev and Render <span class="cite">
	[
	<a href="#AKR2009" >3</a>
	]
</span> introduced a <b class="bf">polynomial</b> Bernstein operator \(B_{n,0,j}: C[0,1]\to \Pi _n\) that fixes \(e_0\) and \(e_j\), and converges in the strong operator topology to the identity as \(n\to \infty \). The operator is a linear combination of the classical Bernstein basis \(\{ p_{n,k}\} _{k=0,\dots ,n}\), thus produces polynomials of degree \(n\) and is explicitly given by </p>
<div class="equation" id="e4">
<p>
  <div class="equation_content">
    \begin{equation} \label{e4} B_{n,0,j}(f;x)=\displaystyle \sum _{k=0}^n f\left(t_{n,k}^j\right)p_{n,k}(x),\end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000007">
  \[ t_{n,k}^j=\left(\tfrac {k(k-1)\dots (k-j+1)}{n(n-1)\dots (n-j+1)}\right)^{1/j}. \]
</div>
<p><div class="remark_thmwrapper " id="a0000000008">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1</span>
  </div>
  <div class="remark_thmcontent">
  <p>(i) For \(j=1\) this is the classical Bernstein operator (reproducing \(e_0\) and \(e_1\)). </p>
<p>(ii) \(B_{n,0,j}\) is linear, positive and of the form </p>
<div class="equation" id="e3">
<p>
  <div class="equation_content">
    \begin{equation}  \label{e3} \displaystyle \sum _{k=0}^nl_{n,k}(f) p_{n,k}(x), \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where \(l_{n,k}\) are positive linear functionals with \( l_{n,k}(e_0)=1\). Obviously all the \(B_{n,0,j}\)s map into \(\Pi _n[0,1]\) (in contrast to the \(B_n^{\tau }\)s from above, case \(\tau = e_j\)). </p>
<p>(iii) For \(j\) fixed and \(n=j\) we have \((t_{n,k}^j)_{k=0,\dots ,n}=(0,\dots ,0,1)\) with \(n\) zeros preceding the \(1\). </p>
<p>For \(n=j+1\) one obtains \((t_{n,k}^j)_{k=0,\dots ,n}=\left(0,\dots ,0,n^{\frac1j},1\right)\), and so on. That is, only for \(n\) big enough all the nodes will be distinct. </p>
<p>(iv) The notation \(B_{n,0,j}\) is motivated by the fact that the operator reproduces \(e_0\) and \(e_j\). As shown by Finta in <span class="cite">
	[
	<a href="#Fin2013" >10</a>
	]
</span>, see Theorem 2.1 there, no sequence \((L_n)\) of type (<a href="#e3">4</a>) can reproduce \(e_i\) and \(e_j\), \(i,j\in \{ 1,2,\dots \} ,\, \,  i{\lt}j\). So reproduction of \(e_0\) is a must! </p>
<p>Moreover, there exist infinitely many sequences of operators \(L_n\) of type (<a href="#e3">4</a>) which approximate each continuous function on \([0, 1]\), and have the functions \(e_0\) and \(e_j\) as fixed points, where \(j\in \{ 1, 2,\dots \} \) is given (see <span class="cite">
	[
	<a href="#Fin2013" >10</a>
	, 
	Th. 2.4
	]
</span>). <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000009">2 On direct estimates for \(B_{{\lowercase {n}},0,{\lowercase {j}}}\)</h1>
<p>We start with some brief history. For the case \(j=2\), \(n\geq 2,\) an inequality involving \(\omega _1\) (following Shisha \(\& \) Mond) was given by Cardenas-Morales <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#CGR2012" >8</a>
	, 
	Proposition 3.1
	]
</span> as follows </p>
<div class="displaymath" id="a0000000010">
  \[ |B_{n,0,2}f(x)-f(x)|\leq \omega _1(f,\delta )\left(1+\tfrac {1}{\delta }\sqrt{2x(1-x)\tfrac {1-(1-x)^{n-1}}{n-1}}\right), \]
</div>
<p> \( f\in C[0,1],\, x\in [0,1]\textrm{ and } \delta {\gt}0. \) </p>
<p>Moreover, in 2014 Finta showed (see <span class="cite">
	[
	<a href="#Fin2014" >11</a>
	]
</span>) </p>
<div class="displaymath" id="a0000000011">
  \[  |B_{n,0,j}(f;x)-f(x)|\leq 2\omega _1\left(f; \left(\tfrac {2j(3+4j)}{\sqrt{n}}\right)^{\frac{1}{2j}}\right), \]
</div>
<p> \( f\in C[0,1],\,  x\in [0,1] \textrm{ and } n\geq j\geq 2.\) </p>
<p>In 2018 Aldaz and Render considered a generalization of the classical Bernstein operators on the polynomial spaces, which reproduce \({\bf 1}\) and a polynomial \(f_1\), strictly increasing on \((0,1)\). Denote by \(B_n^{f_1}\) this Bernstein type operator. For \(f\in C[0,1]\), \(x\in [0,1]\) (see <span class="cite">
	[
	<a href="#AlRe2018" >4</a>
	, 
	Remark following Th. 6.5
	]
</span>) </p>
<div class="displaymath" id="a0000000012">
  \[ | B_n^{f_1}(f;x)-f(x)|\leq (c_S+1)\omega _1\left(f, n^{-\frac{1}{2}}\right),  \]
</div>
<p> where \(c_S=\tfrac {4306+837\sqrt{6}}{5832}\), Sikkema’s constant. The drawback here is that the latter is shown to be valid only for \(n\) sufficiently large. </p>

<p>We will improve the estimates known so far in three different ways. </p>
<p><div class="prop_thmwrapper " id="p2.1">
  <div class="prop_thmheading">
    <span class="prop_thmcaption">
    Proposition
    </span>
    <span class="prop_thmlabel">1</span>
  </div>
  <div class="prop_thmcontent">
  <p> Let \(f\in C[0,1]\), \(x\in [0,1]\), \(n \geq j \geq 1\). Then </p>
<div class="equation" id="D">
<p>
  <div class="equation_content">
    \begin{equation}  \label{D} |f(x)-B_{n,0,j}(f;x)|\leq 1\cdot \omega _2\big(f;\tfrac {1}{\sqrt{n}}\big)+\omega _1\big(f;\tfrac {j-1}{n}\big). \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>

  </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> We have </p>
<div class="displaymath" id="a0000000014">
  \begin{align*} |f(x)-B_{n,0,j}(f;x)| &  \leq |f(x)-B_n(f;x)|+|B_n(f;x)-B_{n,0,j}(f;x)| \\ & \leq 1\cdot \omega _2\big(f;\tfrac {1}{\sqrt{n}}\big)+\omega _1\big(f;\tfrac {j-1}{n}\big). \end{align*}
</div>
<p> The first summand is taken from Păltănea <span class="cite">
	[
	<a href="#Pal2003" >18</a>
	, 
	<a href="#Pal2004" >19</a>
	]
</span>, while the second was given by Acu and Raşa <span class="cite">
	[
	<a href="#AcRa2016" >1</a>
	]
</span>. For \(j=1\) this reduces to a best possible result for classical Bernstein operators. <div class="proof_wrapper" id="a0000000015">
  <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>There is a second way to prove a similar inequality. To this end we use Păltănea’s general theorem for positive linear operators reproducing constant functions (see <span class="cite">
	[
	<a href="#Pal2004" >19</a>
	, 
	Cor.
	
	2.2.1,
	
	p.
	
	31
	]
</span>). </p>
<div class="displaymath" id="a0000000016">
  \begin{align*}  |B_{n,0,j}(f;x)-f(x)|\leq &  \tfrac {1}{h} |B_{n,0,j}(e_1-x;x)|\cdot \omega _1(f;h)\\ & +\left(1+\tfrac {1}{2h^2}B_{n,0,j}\left((e_1-x)^2;x\right)\right)\cdot \omega _2(f;h),\, \,  0{\lt}h\leq \tfrac 12. \end{align*}
</div>
<p><div class="prop_thmwrapper " id="p2.2">
  <div class="prop_thmheading">
    <span class="prop_thmcaption">
    Proposition
    </span>
    <span class="prop_thmlabel">2</span>
  </div>
  <div class="prop_thmcontent">
  <p> Let \(f\in C[0,1]\), \(x\in [0,1]\), \(1 \leq j \leq n\). Then </p>
<ul class="itemize">
  <li><p>\(|B_{n,0,2}(f;x)\! -\! f(x)|\! \leq \!  \tfrac {1}{\sqrt{n-1}} d_n(x)\omega _1\left(f;\tfrac {1}{\sqrt{n\! -\! 1}}\right)\! +\! \left(1\! +\! xd_n(x)\right)\omega _2\left(f;\tfrac {1}{\sqrt{n\! -\! 1}}\right)\), \(n\geq 5\), where \(d_n(x):=(1-x)\left[1-(1-x)^{n-1}\right]\); </p>
</li>
  <li><p>\(|B_{n,0,j}(f;x)-f(x)|\leq (j-1)\tfrac {1}{\sqrt{n}}\omega _1\left(f;\tfrac {1}{\sqrt{n}}\right)+\left(\tfrac {1}{8}+j\right)\omega _2\left(f;\tfrac {1}{\sqrt{n}}\right),\,  n\geq 4. \) </p>
</li>
</ul>

  </div>
</div> <div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> a) In order to use Păltănea’s result we estimate \(B_{n,0,2}(e_1-x;x)\) and<br />\(B_{n,0,2}((e_1-x)^2;x)\). </p>
<p>According to <span class="cite">
	[
	<a href="#CGR2012" >8</a>
	, 
	(3.7)
	]
</span> there holds </p>
<div class="displaymath" id="a0000000018">
  \begin{align*} |B_{n,0,2}(e_1-x;x)|& = |B_{n,0,2}(e_1;x)-x|\\ & \leq \tfrac {1-x}{n-1}\left(1-(1-x)^{n-1}\right) =\tfrac {d_n(x)}{n-1}.\end{align*}
</div>
<p> Furthermore, </p>
<div class="displaymath" id="a0000000019">
  \begin{align*}  0\leq B_{n,0,2}\left((e_1-x)^2;x\right)& =2x\left(x-B_{n,0,2}(e_1;x)\right)\\ & \leq \tfrac {2xd_n(x)}{n-1}. \end{align*}
</div>
<p> Taking \(h=\tfrac {1}{\sqrt{n-1}}\), we get the estimation a). </p>
<p>In order to illustrate the nonsymmetry of the situation, in <a href="#<plasTeX.TeXFragment object at 0x000001CA242DB770>">figure 1</a> are the graphs of the function \(d_n(x)\) for \(n=5\) and \(n=10\), \(x\in [0,1]\). </p>
<figure id="<plasTeX.TeXFragment object at 0x000001CA242DB770>">
  <div class="centered"><img src="img-0001.png" alt="\includegraphics[height=70mm,keepaspectratio]{fig1.png}" style="height:70mm" />
 <figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">\(d_n(x)\) for \(n=5\) and \(n=10\), \(x\in [0,1]\).</span> 
</figcaption>  </div>

</figure>
<p>b) One has to determine global estimates for \(|B_{n,0,j}(e_1-x;x)|\) and <br />\(B_{n,0,j}\left((e_1-x)^2;x\right)\), using again \(B_{n,0,j}(e_0;x)=1\). We have </p>
<div class="displaymath" id="a0000000020">
  \begin{align*}  B_{n,0,j}(e_1-x;x)=\displaystyle \sum _{k=0}^n\left[\left(\tfrac {k}{n}\cdots \tfrac {k-j+1}{n-j+1}\right)^{1/j}-\tfrac {k}{n}\right]p_{n,k}(x). \end{align*}
</div>
<p> The result from <span class="cite">
	[
	<a href="#AcRa2016" >1</a>
	, 
	Section 4.7
	]
</span>, namely </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*} \label{eq.rasa_ana} -\tfrac {j-1}{n}\leq \left(\tfrac {k}{n}\dots \tfrac {k-j+1}{n-j+1}\right)^{1/j}-\tfrac {k}{n}\leq 0, \end{equation*}
</div>
<p> yields </p>
<div class="equation" id="A">
<p>
  <div class="equation_content">
    \begin{equation} \label{A}|B_{n,0,j}(e_1-x;x)|\leq \tfrac {j-1}{n}.\end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> The second moment can be written as </p>
<div class="displaymath" id="B">
  \begin{align} \label{B} 0& \leq B_{n,0,j}\left((e_1-x)^2;x\right)\nonumber \\ & =B_{n,0,j}(e_2;x)-x^2-2x\left[B_{n,0,j}(e_1;x)-x\right]\nonumber \\ & \leq B_{n,0,j}(e_2;x)-x^2+2x\tfrac {j-1}{n}, \end{align}
</div>
<p> where we used (<a href="#A">6</a>). </p>
<p>We have </p>
<div class="displaymath" id="a0000000022">
  \begin{align*} &  B_{n,0,j}(e_2;x)-x^2 \\ & = B_{n,0,j}(e_2;x)-B_n(e_2;x)+B_n(e_2;x)-x^2\\ & =\displaystyle \sum _{k=0}^n\left[\left(\tfrac {k}{n}\cdots \tfrac {k-j+1}{n-j+1}\right)^{\frac{2}{j}}-\left(\tfrac {k}{n}\right)^2\right]p_{n,k}(x)+\tfrac {x(1-x)}{n}\\ & =\displaystyle \sum _{k=0}^n\left[\left(\tfrac {k}{n}\cdots \tfrac {k-j+1}{n-j+1}\right)^{\frac{1}{j}}-\tfrac {k}{n}\right]\left[\left(\tfrac {k}{n}\cdots \tfrac {k-j+1}{n-j+1}\right)^{\frac{1}{j}}+\tfrac {k}{n}\right]p_{n,k}(x)+\tfrac {x(1-x)}{n}\\ & \leq \tfrac {x(1-x)}{n}. \end{align*}
</div>
<p> Together with (<a href="#B">7</a>) this yields </p>
<div class="displaymath" id="a0000000023">
  \[  0\leq B_{n,0,j}\left((e_1-x)^2;x\right)\leq \tfrac {x(1-x)}{n}+2x\tfrac {j-1}{n}.  \]
</div>
<p> So we arrive at </p>
<div class="displaymath" id="C">
  \begin{align} \label{C} |B_{n,0,j}(f;x)-f(x)|& \leq \tfrac {j-1}{n}\tfrac {1}{h}\omega _1(f;h)+\left\{ 1+\tfrac {1}{2h^2}\left[\tfrac {x(1-x)}{n}+2x\tfrac {j-1}{n}\right]\right\} \omega _2(f;h)\nonumber \\ & \leq \tfrac {j-1}{n}\tfrac {1}{h}\omega _1(f;h)+\left\{ 1+\tfrac {1}{2nh^2}\left(\tfrac {1}{4}+2j\right)\right\} \omega _2(f;h). \end{align}
</div>
<p> Let \(h=\tfrac {1}{\sqrt{n}}\), \(n\geq 4\). This leads to our proposition. </p>
<p><div class="proof_wrapper" id="a0000000024">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000025">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2</span>
  </div>
  <div class="remark_thmcontent">
  <p>a) The right hand side of inequality (<a href="#C">8</a>) is not a pointwise one and so does not express interpolation at the endpoints. There is room for improvement. </p>
<p>b) If one compares the estimate from <a href="#p2.2">proposition 2</a> b) and the estimate from <a href="#p2.1">proposition 1</a> it is clear that \(\omega _2\big(f;\tfrac {1}{\sqrt{n}}\big)\) is always the worse (<i class="it">i.e.</i>, dominant) term. In case \(f\in C^2[0,1]\) both inequalities give \({\mathcal O}\big(\tfrac {1}{n}\big)\). This cannot be reached by any of the previous inequalities in terms of \(\omega _1\) only. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>The inequalities (<a href="#D">5</a>) and those from <a href="#p2.2">proposition 2</a> do not imply that the function \(e_j\) is reproduced by \(B_{n,0,j}\) if \(j\geq 2\). In order to achieve this we follow an idea of Marsden and Schoenberg <span class="cite">
	[
	<a href="#Schoenberg" >17</a>
	, 
	p.
	
	82
	]
</span> (see also <span class="cite">
	[
	<a href="#ANTA" >14</a>
	, 
	Corollary 3.4
	]
</span>). Below we will use a variation of an analogue of their \(\omega _1^*(f;\delta )\), namely \( \omega _j^*(f;\delta )\) given by </p>
<div class="displaymath" id="a0000000026">
  \[  \omega _j^*(f;\delta ):=\displaystyle \inf _{l}\left\{ \omega _1(f-l;\delta )\right\} ,\textrm{ where } l(x)=ae_j(x),\, \,  a \in \mathbb {R}.  \]
</div>
<p> Note that we won’t loose anything in doing so! </p>
<p><div class="prop_thmwrapper " id="p2.3">
  <div class="prop_thmheading">
    <span class="prop_thmcaption">
    Proposition
    </span>
    <span class="prop_thmlabel">3</span>
  </div>
  <div class="prop_thmcontent">
  <p> For \(f\in C[0,1]\), \(x\in [0,1]\) and \(n \geq j\) we have </p>
<div class="displaymath" id="a0000000027">
  \begin{align*}  |f(x)-B_{n,0,j}(f;x)|\leq \displaystyle \inf _{l}\left\{ \omega _2\big(f-l;\tfrac {1}{\sqrt{n}}\big)+\omega _1\big(f-l;\tfrac {j-1}{n}\big)\right\} . \end{align*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000028">
  <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 \(B_{n,0,j}\) reproduces \(e_j\) we have </p>
<div class="displaymath" id="a0000000029">
  \begin{align*}  |f(x)-B_{n,0,j}(f;x)|& =\left|(f-l)(x)-B_{n,0,j}(f-l;x)\right|\\ & \leq 1\cdot \omega _2\left(f-l;\tfrac {1}{\sqrt{n}}\right)+\omega _1\left(f-l;\tfrac {j-1}{n}\right). \end{align*}
</div>
<p> Since \(l=ae_j\) was arbitrary, we pass to the \(inf\) and get the result. </p>
<p>Moreover, if \(f\) is of the form \(f(x)=a_0+a_j x^j,\,  a_0,a_j\in {\mathbb R}\), we arrive at </p>
<div class="displaymath" id="a0000000030">
  \[ |f(x)-B_{n,0,j}(f;x)|=0.  \]
</div>
<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> Analogous statements follow from the inequalities in <a href="#p2.2">proposition 2</a>. </p>
<p><div class="remark_thmwrapper " id="a0000000032">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3</span>
  </div>
  <div class="remark_thmcontent">
  <p>Except for the papers previously cited we did not find any reference mentioning or using \(\omega _l^*\) or modifications thereof. The similarity to K-functionals is obvious. </p>
<p>In <span class="cite">
	[
	<a href="#ANTA" >14</a>
	]
</span> it is mentioned that </p>
<div class="displaymath" id="a0000000033">
  \[  \omega _1^*(f;\delta ) \leq c \cdot \omega _2(f;\sqrt\delta ).  \]
</div>

  </div>
</div> </p>
<h1 id="a0000000034">3 Iterates of \( B_{{\lowercase {n}},0,{\lowercase {j}}}\)</h1>
<p> Using the approach of Agratini and Rus (see <span class="cite">
	[
	<a href="#AgRu2003" >2</a>
	]
</span>) the following result for the sequence of iterates \((B_{n,0,j}^m)_{m\geq 1}\) is obtained. <div class="theorem_thmwrapper " id="a0000000035">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For \(n\geq j\) fixed and \(m\geq 1\), the iterates sequence \((B_{n,0,j}^m)_{m\geq 1}\) satifies </p>
<div class="displaymath" id="a0000000036">
  \[ \displaystyle \lim _{m\to \infty }B_{n,0,j}^m(f;x)=f(0)+\left[f(1)-f(0)\right]e_j(x). \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000037">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Denote </p>
<div class="displaymath" id="a0000000038">
  \[  X_{\alpha ,\beta }:=\left\{ f\in C[0,1]\, \, \left|\, \,  f(0)=\alpha ,\, f(1)=\beta \right.\right\} , \, \,  (\alpha ,\beta )\in {\mathbb R}\times {\mathbb \mathbb {R}}.  \]
</div>
<p> The operators \(B_{n,0,j}\) interpolate \(0\) and \(1\): </p>
<div class="displaymath" id="a0000000039">
  \begin{align*} & B_{n,0,j}(f;0)=f(t_{n,0}^j)p_{n,0}(x)=f(0),\\ & B_{n,0,j}(f;1)=f(t_{n,n}^j)p_{n,0}(x)=f(1). \end{align*}
</div>
<p> Let \(f,g\in X_{\alpha ,\beta }\). Then, for \(x\in [0,1]\), </p>
<div class="displaymath" id="a0000000040">
  \begin{align*}  |B_{n,0,j}(f;x)-B_{n,0,j}(g;x)|& =\left|\displaystyle \sum _{k=1}^{n-1}[f(t_{n,k})-g(t_{n,k})]p_{n,k}(x)\right|\\ & \leq \left(1-p_{n,0}(x)-p_{n,n}(x)\right)\cdot \| f-g\| _{\infty }\\ & \leq \left(1-\tfrac {1}{2^{n-1}}\right)\| f-g\| _{\infty }. \end{align*}
</div>
<p> Therefore, \(B_{n,0,j}|_{X_{\alpha ,\beta }}:X_{\alpha ,\beta }\to X_{\alpha ,\beta } \) is a contraction for every \((\alpha , \beta )\in {\mathbb R}\times {\mathbb R}\) and \(n\geq j\) fixed. </p>
<p>The function \(p_{\alpha ,\beta }^*:=f(0)+\left(f(1)-f(0)\right)e_j\) belongs to \(X_{f(0),f(1)}\) and is a fixed point of \(B_{n,0,j}\). </p>
<p>By contradiction we obtain the claim. <div class="proof_wrapper" id="a0000000041">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>The above result was also obtained by Gavrea and Ivan as an example for a general theorem concerning the limit of the iterates of positive linear operators (see <span class="cite">
	[
	<a href="#GaIv2011" >13</a>
	]
</span>). No corresponding quantitative version is known to us. </p>
<h1 id="a0000000042">4 Voronovskaya result for the Bernstein-type operators \(B_{{\lowercase {n}},0,{\lowercase {j}}}\)</h1>
<p> The asymptotic formula of the operator \(B_{n,0,j}\) was stated as a conjecture by Cárdenas-Morales <i class="it">et al.</i> <span class="cite">
	[
	<a href="#CGR2012" >8</a>
	]
</span> and proved by Birou in <span class="cite">
	[
	<a href="#Bir2017" >5</a>
	]
</span>. </p>
<p><div class="theorem_thmwrapper " id="a0000000043">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
    <span class="theorem_thmtitle">see <span class="cite">
	[
	<a href="#CGR2012" >8</a>
	]
</span>, <span class="cite">
	[
	<a href="#Bir2017" >5</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> For all \(f\in C[0,1]\), \(x\in (0,1)\) and \(j\geq 1\), whenever \(f^{\prime \prime }(x)\) exists, </p>
<div class="equation" id="e2v">
<p>
  <div class="equation_content">
    \begin{equation}  \label{e2v} \displaystyle \lim _{n\to \infty } n(B_{n,0,j}(f;x)-f(x))=\tfrac {x(1-x)}{2}f^{\prime \prime }(x)-\tfrac {(j-1)(1-x)}{2}f^{\prime }(x). \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>

  </div>
</div> </p>
<p>A quantitative <b class="bf">pre-Voronovskaya theorem</b> for the Bernstein type operator \(B_{n,0,j}\) was proved by Finta <span class="cite">
	[
	<a href="#Finta1" >12</a>
	]
</span> using the first order Ditzian-Totik modulus of smoothness defined by </p>
<div class="displaymath" id="a0000000044">
  \[  \omega _{\varphi }^1(f;\delta )=\displaystyle \sup _{0{\lt}h\leq \delta }\sup _{x\pm \frac{1}{2}h\varphi (x)}\left|f\big(x+\tfrac {1}{2}h\varphi (x)\big)-f\big(x-\tfrac {1}{2}h\varphi (x)\big)\right|,  \]
</div>
<p> where \(\varphi (x)=\sqrt{x(1-x)}\), \(x\in [0,1]\). We call this "pre-Voronovskaya" since it appears to point into a good direction but the limit is not explicitly given. <div class="theorem_thmwrapper " id="a0000000045">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p><span class="cite">
	[
	<a href="#Finta1" >12</a>
	]
</span> There exists \(C{\gt}0\) depending only on \(j\) such that </p>
<div class="displaymath" id="a0000000046">
  \begin{align*}  \Big|n(B_{n,0,j}(f;x)-f(x))+ &  f^{\prime }(x)nB_{n,0,j}(xe_0-e_1;x) \\ & -\tfrac {1}{2} f^{\prime \prime }(x)nB_{n,0,j}((e_1-xe_0)^2;x)\Big| \leq C\omega _{\varphi }^1\big(f^{\prime \prime };\tfrac {1}{\sqrt{n}}\big), \end{align*}
</div>
<p> for all \(x\in [0,1]\), \(f\in C^2[0,1]\) and \(n\geq j \geq 1\). </p>

  </div>
</div> </p>
<p>No quantitative version of a Voronovskaya-type theorem for the operators of Aldaz <i class="it">et al.</i> is known to us. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="AcRa2016">1</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11075-016-0117-8"> <i class="sc">A.M. Acu, I. Raşa</i>, <i class="it">New estimates for the differences of positive linear operators</i>, Numer. Algor., <b class="bf">73</b> (2016) no. 3, pp. 775–789, <a href="https://doi.org/10.1007/s11075-016-0117-8">https://doi.org/10.1007/s11075-016-0117-8</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AgRu2003">2</a></dt>
  <dd><p><i class="sc">O. Agratini, I.A. Rus</i>, <i class="it">Iterates of a class of discrete linear operator via contraction principle</i>, Comment. Math. Univ. Carol., <b class="bf">44</b> (2003) 3, 555–563. </p>
</dd>
  <dt><a name="AKR2009">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00211-009-0248-0"> <i class="sc">J.M. Aldaz, O. Kounchev, H. Render</i>, <i class="it">Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces</i>, Numer. Math., <b class="bf">114</b> (2009) no. 1, pp. 1–25, <a href="https://doi.org/10.1007/s00211-009-0248-0">https://doi.org/10.1007/s00211-009-0248-0</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AlRe2018">4</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00009-018-1266-x"> <i class="sc">J.M. Aldaz, H. Render</i>, <i class="it">Generalized Bernstein operators on the classical polynomial spaces</i>, Mediterr. J. Math., <b class="bf">15</b> (2018) art. id. 222, <a href="https://doi.org/10.1007/s00009-018-1266-x">https://doi.org/10.1007/s00009-018-1266-x</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Bir2017">5</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00025-016-0608-x"> <i class="sc">M. Birou</i>, <i class="it">A proof of a conjecture about the asymptotic formula of a Bernstein type operator</i>, Results Math., <b class="bf">72</b> (2017), pp. 1129–1138, <a href="https://doi.org/10.1007/s00025-016-0608-x">https://doi.org/10.1007/s00025-016-0608-x</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="CGM2006">6</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.amc.2006.05.046"> <i class="sc">D. Cárdenas-Morales, P. Garrancho, F.J. Muñoz-Delgado</i>, <i class="it">Shape preserving approximation by Bernstein-type operators which fix polynomials</i>, Appl. Math. Comput., <b class="bf">182</b> (2006), 1615–1622, <a href="https://doi.org/10.1016/j.amc.2006.05.046">https://doi.org/10.1016/j.amc.2006.05.046</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="CGR2011">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.camwa.2011.04.063"><i class="sc">D. Cárdenas-Morales, P. Garrancho, I. Raşa</i>, <i class="it">Bernstein-type operators which preserve polynomials</i>, Comput. Math. Appl., <b class="bf">62</b> (2011), 158–163, <a href="https://doi.org/10.1016/j.camwa.2011.04.063">https://doi.org/10.1016/j.camwa.2011.04.063</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="CGR2012">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1155/2012/217464"> <i class="sc">D. Cárdenas-Morales, P. Garrancho, I. Raşa</i>, <i class="it">Asymptotic formulae via a Korovkin-type result</i>, Abstract Appl. Anal., <b class="bf">2012</b> (2021), art. ID 217464, 12 pp., <a href="https://doi.org/10.1155/2012/217464">https://doi.org/10.1155/2012/217464</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="CGGKZ1999">9</a></dt>
  <dd><p><a href =" https://doi.org/10.1155/s1025583499000314"> <i class="sc">C. Cottin, I. Gavrea, H. Gonska, D. Kacsó, Ding-Xuan Zhou</i>, <i class="it">Global smoothness preservation and the variation-diminishing property</i>, J. Inequal. Appl., <b class="bf">4</b> (1999) no. 2, 91–114, <a href="https://doi.org/10.1155/s1025583499000314">https://doi.org/10.1155/s1025583499000314</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Fin2013">10</a></dt>
  <dd><p><i class="sc">Z. Finta</i>, <i class="it">Bernstein type operators having \(1\) and \(x^j\) as fixed points</i>, Cent. Eur. J. Math., <b class="bf">11</b> (2013), 2257–2261. </p>
</dd>
  <dt><a name="Fin2014">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.jmaa.2014.02.010"> <i class="sc">Z. Finta</i>, <i class="it">Note on a Korovkin-type theorem</i>, J. Math. Anal. Appl., 415 (2014), 750–759, <a href="https://doi.org/10.1016/j.jmaa.2014.02.010">https://doi.org/10.1016/j.jmaa.2014.02.010</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Finta1">12</a></dt>
  <dd><p><a href ="https://doi.org/10.33205/cma.553427"> <i class="sc">Z. Finta</i>, <i class="it">A quantitative variant of Voronovskaja’s theorem for King-type operators</i>, Constructive Mathematical Analysis, <b class="bf">2</b> (2019) no. 3, 124–129, <a href="https://doi.org/10.33205/cma.553427">https://doi.org/10.33205/cma.553427</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="GaIv2011">13</a></dt>
  <dd><p><a href ="https://doi.org/10.1155/2011/670509"> <i class="sc">I. Gavrea, M. Ivan</i>, <i class="it">Asymptotic behaviour of the iterates of positive linear operators</i>, Abstract Appl. Anal., <b class="bf">2011</b> (2011), art. ID 670509, 11 pp., <a href="https://doi.org/10.1155/2011/670509">https://doi.org/10.1155/2011/670509</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="ANTA">14</a></dt>
  <dd><p><a href ="https://ictp.acad.ro/jnaat/journal/article/view/1985-vol14-no1-art2"> <i class="sc">H. Gonska</i>, <i class="it">On approximation by linear operators: improved estimates</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">14</b> (1985), 7–32, <a href="https://ictp.acad.ro/jnaat/journal/article/view/1985-vol14-no1-art2">https://ictp.acad.ro/jnaat/journal/article/view/1985-vol14-no1-art2</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="GoPi2005">15</a></dt>
  <dd><p><i class="sc">H. Gonska, P. Piţul</i>, <i class="it">Remarks on an article of J.P. King</i>, Comment. Math. Univ. Carol., <b class="bf">46</b> (2005), 645–652. </p>
</dd>
  <dt><a name="1">16</a></dt>
  <dd><p><a href ="https://doi.org/10.1023/a:1024571126455"> <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> (2003) no. 3, 203–208, <a href="https://doi.org/10.1023/a:1024571126455">https://doi.org/10.1023/a:1024571126455</a>. <img src="img-0002.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Schoenberg">17</a></dt>
  <dd><p><i class="sc">M. Marsden, I.J. Schoenberg</i>, <i class="it">On variation diminishing spline approximation methods</i>, Mathematica, <b class="bf">8 (31)</b> (1966) no. 1, 61–82. </p>
</dd>
  <dt><a name="Pal2003">18</a></dt>
  <dd><p><i class="sc">R. Păltănea</i>, <i class="it">Optimal constant in approximation by Bernstein operators</i>, J. Comput. Anal. Appl., <b class="bf">5</b> (2003), 195–235. </p>
</dd>
  <dt><a name="Pal2004">19</a></dt>
  <dd><p><i class="sc">R. Păltănea</i>, <i class="it">Approximation Theory using Positive Linear Operators</i>, Birkhäuser, Boston, 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>