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<!DOCTYPE html><html  xmlns:dc="http://purl.org/dc/terms/" xmlns:og="http://ogp.me/ns#" ><head><meta http-equiv=Content-Type content="text/html; charset=utf-8"><title>Notes on Approximation Theory</title><meta name="citation_pdf_url" content="https://peteroupc.github.io/approxtheory.pdf"><meta name="citation_url" content="https://peteroupc.github.io/approxtheory.html"><meta name="citation_title" content="Notes on Approximation Theory"><meta name="dc.date" content="2026-05-24"><meta name="citation_date" content="2026/05/24"><meta name="citation_publication_date" content="2026/05/24"><meta name="citation_online_date" content="2026/05/24"><meta name="og:title" content="Notes on Approximation Theory"><meta name="og:type" content="article"><meta name="og:url" content="https://peteroupc.github.io/approxtheory.html"><meta name="og:site_name" content="peteroupc.github.io"><meta name="dc.format" content="text/html"><meta name="dc.language" content="en"><meta name="title" content="Notes on Approximation Theory"><meta name="dc.title" content="Notes on Approximation Theory"><meta name="twitter:title" content="Notes on Approximation Theory"><meta name="dc.creator" content="Peter Occil"/><meta name="author" content="Peter Occil"/><meta name="citation_author" content="Peter Occil"/><meta name="viewport" content="width=device-width"><link rel=stylesheet type="text/css" href="/style.css">
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<h1 id="notes-on-approximation-theory">Notes on Approximation Theory</h1><p>This version of the document is dated 2026-05-24.  <a href='https://github.com/peteroupc/peteroupc.github.io/issues'>Post an issue or comment on this document</a>.</p>

<p><a href="mailto:poccil14@gmail.com"><strong>Peter Occil</strong></a></p>

<p>Some notes that may be useful when finding approximation error bounds that are explicit, with no hidden constants and without introducing transcendental or trigonometric functions.</p>

<p>The notes generally relate to finding bounds on how close a polynomial is to a single-variable function on a closed interval.  The mapping from a function to a function (in this case, from a single-variable function to a polynomial “close” to it) is called an <em>operator</em>, and operators involved in these bounds are often linear operators, whose behavior is relatively simple to examine.</p>

<p><a id="Contents"></a></p>

<h2 id="contents">Contents</h2>

<ul>
  <li><a href="#Contents"><strong>Contents</strong></a></li>
  <li><a href="#Notation_and_Definitions"><strong>Notation and Definitions</strong></a></li>
  <li><a href="#Bernstein_Form_and_Bernstein_Polynomials"><strong>Bernstein Form and Bernstein Polynomials</strong></a></li>
  <li><a href="#Moments_of_Linear_Operators"><strong>“Moments” of Linear Operators</strong></a>
    <ul>
      <li><a href="#Moments_of_Bernstein_Polynomials"><strong>“Moments” of Bernstein Polynomials</strong></a></li>
    </ul>
  </li>
  <li><a href="#Taylor_Expansion_of_Linear_Operators"><strong>Taylor Expansion of Linear Operators</strong></a></li>
  <li><a href="#Results_on_Error_Bounds"><strong>Results on Error Bounds</strong></a>
    <ul>
      <li><a href="#Bounds_for_General_Positive_Linear_Operators"><strong>Bounds for General Positive Linear Operators</strong></a></li>
      <li><a href="#Bounds_for_Remainder_of_Bernstein_Polynomials"><strong>Bounds for Remainder of Bernstein Polynomials</strong></a></li>
      <li><a href="#Bounds_for_General_Linear_Operators"><strong>Bounds for General Linear Operators</strong></a></li>
      <li><a href="#Whitney_s_Inequality_on_Polynomial_Errors"><strong>Whitney’s Inequality on Polynomial Errors</strong></a></li>
      <li><a href="#Another_Inequality_on_Polynomial_Errors"><strong>Another Inequality on Polynomial Errors</strong></a></li>
      <li><a href="#Lebesgue_Inequality_for_Certain_Linear_Operators"><strong>Lebesgue Inequality for Certain Linear Operators</strong></a></li>
      <li><a href="#Bounds_for_Certain_Nonlinear_Operators"><strong>Bounds for Certain Nonlinear Operators</strong></a></li>
    </ul>
  </li>
  <li><a href="#Example"><strong>Example</strong></a></li>
  <li><a href="#Example_An_Interesting_Linear_Operator"><strong>Example: An Interesting Linear Operator</strong></a></li>
  <li><a href="#Probabilistic_Interpretations_of_Linear_Operators"><strong>Probabilistic Interpretations of Linear Operators</strong></a></li>
  <li><a href="#License"><strong>License</strong></a></li>
  <li><a href="#Notes"><strong>Notes</strong></a></li>
</ul>

<p><a id="Notation_and_Definitions"></a></p>

<h2 id="notation-and-definitions">Notation and Definitions</h2>

<p>For definitions of <em>continuous</em>, <em>derivative</em>, <em>convex</em>, <em>concave</em>, <em>bounded</em>, <em>Hölder continuous</em>, and <em>Lipschitz continuous</em>, see the definitions section in “<a href="https://peteroupc.github.io/bernsupp.html#Definitions"><strong>Supplemental Notes for Bernoulli Factory Algorithms</strong></a>”.</p>

<ul>
  <li>The <em>closed unit interval</em> (written as [0, 1]) means the set consisting of 0, 1, and every real number in between.</li>
  <li>An <em>operator</em> is a mapping from a function to a function.</li>
  <li>An operator $L$ is <em>linear</em> if it satisfies $L(af)=aL(f)$ and $L(f+g)=L(f)+L(g)$ for all allowed functions $f$ and $g$ and every number $a$.</li>
  <li>An operator $L$ is <em>positive</em> if it has the property that, if an allowed function $f$ is nonnegative on its domain, so is $L(f)$.<sup id="fnref:1"><a href="#fn:1" class="footnote" rel="footnote" role="doc-noteref">1</a></sup></li>
  <li>The <em>operator norm</em> of an operator $L$ is the maximum absolute value of $L(f)$ over all allowed functions $f$ with a maximum absolute value 1 or less.  This assumes $L$ maps continuous functions on a closed interval to functions of that kind.</li>
  <li>In this document, $e_i$ is a function such that $e_i(t) = t^i$, so that $e_0(t) = 1$ and $e_1(t) = t$; as an example, if $L(f) = f(0) + f(1)$, then $L(e_1 - x)$ = $(e_1(0) - x) + (e_1(1) - x)$ = $(0-x)+(1-x)=1-2x$.</li>
  <li>The <em>expected value</em> (or mean or “long-run average”) of a random variable $Y$ is denoted $\mathbb{E}[Y]$.</li>
  <li>A <em>modulus of continuity of order 1</em> of a function <em>f</em>, denoted $\omega_1(f, \delta)$, means a nonnegative and nowhere decreasing function where, for each $\delta\ge 0$, $\text{abs}(f(x)-f(y))\le\omega_1(f, \delta)$ whenever $x$ and $y$ are in $f$’s domain and no more than $\delta$ apart.  Loosely speaking, $\omega_1(f, \delta)$ gives how much $f$ can vary when $f$ is restricted to a window of size $\delta$ or less.  The modulus of continuity reflects the “regularity” of $f$; generally, the smaller it is, the more “regular”.</li>
</ul>

<p><a id="Bernstein_Form_and_Bernstein_Polynomials"></a></p>

<h2 id="bernstein-form-and-bernstein-polynomials">Bernstein Form and Bernstein Polynomials</h2>

<p>Among the best known examples of linear operators are the Bernstein polynomials.</p>

<p>In this document, a polynomial $P(x)$ is written in <em>Bernstein form of degree $n$</em> if it is written as—</p>

\[P(x)=\sum_{k=0}^n a_k \frac{n!}{(k!)((n-k)!)} x^k (1-x)^{n-k},\]

<p>where the real numbers $a_0, …, a_n$ are the polynomial’s <em>Bernstein coefficients</em>.<sup id="fnref:2"><a href="#fn:2" class="footnote" rel="footnote" role="doc-noteref">2</a></sup></p>

<p>The degree-$n$ <em>Bernstein polynomial</em> of an arbitrary function $f(x)$ has Bernstein coefficients $a_k = f(k/n)$.  In general, this Bernstein polynomial differs from $f$ even if $f$ is a polynomial.  In this document, the degree-$n$ Bernstein polynomial of $f$ is denoted $B_n(f)$. $B_n(f)$ is a positive linear operator.</p>

<p><a id="Moments_of_Linear_Operators"></a></p>

<h2 id="moments-of-linear-operators">“Moments” of Linear Operators</h2>

<p>To examine the approximation behavior of linear operators, it is helpful to find the so-called “moments” of those operators, that is, the functions they map certain functions to.</p>

<p>For a linear operator $L$, they are:</p>

<ul>
  <li>“Raw moments”: The values of $L(e_i)$ for each integer $i\ge 0$.</li>
  <li>“Central moments”: The values of $L((e_1-x)^i)$ for each integer $i\ge 0$.  If the “raw moments” $L(e_0), …, L(e_j)$ are known, then $L((e_1-x)^j)$ is also known, thanks to proposition 5.6 of Gonska et al. (2006)<sup id="fnref:3"><a href="#fn:3" class="footnote" rel="footnote" role="doc-noteref">3</a></sup>.</li>
  <li>“Absolute moments”: The values of $L(\text{abs}(e_1-x)^i)(x)$ for each integer $i\ge 0$.  When $i$ is even, $L(\text{abs}(e_1-x)^i)$ = $L((e_1-x)^i)$.</li>
</ul>

<p>Because $L$ is linear, if $L(e_i) = e_i$ for each $i$ from 0 through $j$ ($j$ is zero or a positive integer), then:</p>

<ul>
  <li>$L$ <em>reproduces all polynomials</em> up to degree $j$ (that is, $L(f) = f$ whenever $f$ is a polynomial of degree $j$ or less).</li>
  <li>The $0$-th “central moment” is $L(e_0)$ = $L((e_1-x)^0)$ = 1, and for each $i$ from 1 through $j$, $L((e_1-x)^j) = 0$.</li>
</ul>

<p>Also, because $L$ is linear, the “moments” of degree up to $m$, say, lead to easy ways to find the mapping by $L$ of any polynomial of degree up to $m$, when the polynomial is written in “power” form.</p>

<blockquote>
  <p><strong>Example:</strong> Let $f(x)$ be the polynomial $4x^3 - 6x^2 + 8x^1 - 10$.  Then:</p>

\[L(f) = 4L(e_3) - 6L(e_2) + 8L(e_1) - 10L(e_0).\]
</blockquote>

<p><a id="Moments_of_Bernstein_Polynomials"></a></p>

<h3 id="moments-of-bernstein-polynomials">“Moments” of Bernstein Polynomials</h3>

<p>The following results deal with useful quantities when discussing the error in approximating a function by Bernstein polynomials.</p>

<p>Suppose a coin shows heads with probability $p$, and $n$ independent tosses of the coin are made, where $n$ is 1 or greater.  Then the total number of heads $X$ follows a <em>binomial distribution</em>.  The following are useful quantities of this distribution.</p>

<ul>
  <li>$T_{n,r}$: The <em>central moment</em> (moment about the mean) of $X$ is denoted $T_{n,r}(p)$ = $\mathbb{E}[(X-\mathbb{E}[X])^r]$ = $B_n((e_1-p)^r)(p)\cdot n^r$. Formulas for computing this central moment are given in Skorski (2024)<sup id="fnref:4"><a href="#fn:4" class="footnote" rel="footnote" role="doc-noteref">4</a></sup>.</li>
  <li>$S_{n,r}$: Traditionally, the central moment of $X/n$ or the ratio of heads to tosses is denoted $S_{n,r}(p)$ = $T_{n,r}(p)/n^r$ = $\mathbb{E}[(X/n-\mathbb{E}[X/n])^r]$ = $B_n((e_1-p)^r)(p)$.  ($T$ and $S$ are notations of S.N. Bernstein, known for Bernstein polynomials.) $S_{n,r}$ is thus the $r$-th “central moment” of degree-$n$ Bernstein polynomials.</li>
  <li>$M_{n,r}$: The $r$-th <em>central absolute moment</em> of $X/n$ is denoted $M_{n,r}(p)$ = $\mathbb{E}[\text{abs}(X/n-\mathbb{E}[X/n])^r]$ = $B_n(\text{abs}(e_1-p)^r)(p)$.  If $r$ is even, $M_{n,r}(p) = S_{n,r}(p)$. $M_{n,r}$ is thus the $r$-th “absolute moment” of degree-$n$ Bernstein polynomials.</li>
</ul>

<p>The following gives bounds on $M_{n,r}$; some results in approximation theory rely on bounds like these.<sup id="fnref:5"><a href="#fn:5" class="footnote" rel="footnote" role="doc-noteref">5</a></sup></p>

<p><strong>Proposition 1</strong>: <em>Let $r\ge 0$, and let $\sigma(r,t) = (r!)/(((r/2)!)t^{r/2})$.  Then for real numbers $r$ and integers $n$ described in the following table,</em> $M_{n, r}(p)\le \mu_{n,r}/n^{r/2}$, <em>where</em> $\mu_{n,r}$ <em>is as given in the table.</em></p>

<table>
  <thead>
    <tr>
      <th>If $r$…</th>
      <th>Then $\mu_{n,r}$ is…</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Is an even integer.</td>
      <td>$\sigma(r,6)$, for positive $n$.</td>
    </tr>
    <tr>
      <td>Is an even integer, but not greater than 44.</td>
      <td>$\sigma(r,8)$, for positive $n$.</td>
    </tr>
    <tr>
      <td>Is 1.</td>
      <td>$1/2$, for positive $n$.</td>
    </tr>
    <tr>
      <td>Is odd, and $3\le r\le 43$.</td>
      <td>$\sqrt{\sigma(r-1,8)\sigma(r+1,8)} = r^{1/2}(r-1)! / (2\cdot 8^{(r-1)/2}((r-1)/2)!)$, for $n\ge 2$.</td>
    </tr>
    <tr>
      <td>Is odd and greater than 43.</td>
      <td>$\sqrt{\sigma(r-1,6)\sigma(r+1,6)}$, for $n\ge 2$.</td>
    </tr>
  </tbody>
</table>

<p><em>Proof:</em> The first row comes from a result of Adell and Cárdenas-Morales (2018)<sup id="fnref:6"><a href="#fn:6" class="footnote" rel="footnote" role="doc-noteref">6</a></sup>.  The second row is an improved result of the first, from Molteni (2022)<sup id="fnref:7"><a href="#fn:7" class="footnote" rel="footnote" role="doc-noteref">7</a></sup>.  The third row follows from Cheng (1983)<sup id="fnref:8"><a href="#fn:8" class="footnote" rel="footnote" role="doc-noteref">8</a></sup>.  The fourth and fifth rows follow from the first and second as well as that the absolute central moment for odd $r$ can be bounded for every integer $n\ge 2$, using Schwarz’s inequality (Weisstein)<sup id="fnref:9"><a href="#fn:9" class="footnote" rel="footnote" role="doc-noteref">9</a></sup> (see also Bojanić and Shisha 1975<sup id="fnref:10"><a href="#fn:10" class="footnote" rel="footnote" role="doc-noteref">10</a></sup> for the case $r=4$). □</p>

<p><a id="Taylor_Expansion_of_Linear_Operators"></a></p>

<h2 id="taylor-expansion-of-linear-operators">Taylor Expansion of Linear Operators</h2>

<p>Continuous functions can be “unwrapped” into a Taylor expansion.  The linear mapping of those functions also has a Taylor expansion of sorts, which is described next.</p>

<p>Let $f(\lambda)$ have a continuous $s$-th derivative on a closed interval, where $s$ is zero or a positive integer, and let $L(f)$ be a linear operator that maps continuous functions on that interval to functions of that kind.  Then:</p>

\[L(f)(\lambda) = L(R_s(f, \lambda)) + \sum_{i=0}^s L((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!}, \tag{1}\]

<p>where $R_s(f,\lambda)$ is the remainder after subtracting from $f$ the degree-$s$ Taylor polynomial of $f$ centered at $\lambda$. (See also Piţul (2007, proof of theorem 5.8)<sup id="fnref:11"><a href="#fn:11" class="footnote" rel="footnote" role="doc-noteref">11</a></sup>.)  $R_s(f,\lambda)$ is 0 if $f$ is a polynomial of degree $s$ or less.</p>

<p>If $L$ reproduces constants, so that $L(e_0)=1$, this becomes:</p>

\[L(f)(\lambda) - f(\lambda) = L(R_s(f, \lambda)) + \sum_{i=1}^s L((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!}.\tag{2}\]

<p>If $L$ reproduces polynomials up to degree $s$, this even reduces to $L(f)(\lambda) - f(\lambda) = L(R_s(f, \lambda))$.</p>

<p>It can be seen from the expansions just given that finding upper bounds for $L_n(f)(\lambda)$ involves:</p>

<ul>
  <li>Finding upper bounds for $L$’s “central moments” up to the $s$-th order.</li>
  <li>Finding upper bounds for $L(R_s(f,\lambda))$. If $L$ is positive linear, such bounds are given in the section “<a href="#Bounds_for_General_Positive_Linear_Operators"><strong>Bounds for General Positive Linear Operators</strong></a>”. If $L$ is nonpositive linear, bounds are given in the section “<a href="#Bounds_for_General_Linear_Operators"><strong>Bounds for General Linear Operators</strong></a>”, and this can be helped if $L$ can be written as a difference between two positive linear operators $LA$ and $LB$, so that $L(f) = LA(f) - LB(f)$.<sup id="fnref:12"><a href="#fn:12" class="footnote" rel="footnote" role="doc-noteref">12</a></sup>  See the “<a href="#Example"><strong>Example</strong></a>” section later in this document.</li>
</ul>

<p>Meanwhile, bounds for the derivatives of $f$ (here, $f^{(i)}$) are often assumed to be known beforehand.</p>

<p><a id="Results_on_Error_Bounds"></a></p>

<h2 id="results-on-error-bounds">Results on Error Bounds</h2>

<p>Some results on error bounds for certain classes of operators.</p>

<p>In this section, $\Vert g\Vert _\infty$ is the maximum of the absolute value of (the continuous function) $g$ on its domain.</p>

<p><a id="Bounds_for_General_Positive_Linear_Operators"></a></p>

<h3 id="bounds-for-general-positive-linear-operators">Bounds for General Positive Linear Operators</h3>

<p>The following results give bounds that apply to large classes of positive linear operators.  In this section:</p>

<ul>
  <li>$\sigma_i = L((e_i-\lambda)^i)(\lambda)$ (the $i$-th “central moment” of the linear operator $L$ in question).</li>
  <li>$\tau_i = L(\text{abs}(e_i-\lambda)^i)(\lambda)$  (the $i$-th “absolute moment” of the linear operator $L$ in question).</li>
  <li>$\omega_1(f, \delta)$ is the smallest modulus of continuity of a function $f$ of order 1, with parameter $\delta$.</li>
  <li>$\tilde\omega_1(f, \delta)$ is the smallest concave modulus of continuity of $f$ of order 1, both with parameter $\delta$.</li>
</ul>

<p><strong>Lemma 1</strong>. <em>Let $f(\lambda)$ be continuous on a closed interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind and reproduces all constants (so that</em> $L(e_0) = 1$ <em>).  Then:</em></p>

<table>
  <thead>
    <tr>
      <th>No.</th>
      <th>$\text{abs}(L(f)(\lambda)-f(\lambda))\le …$</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>1</td>
      <td>$\tilde\omega_1(f, \tau_1)$.</td>
    </tr>
    <tr>
      <td>2</td>
      <td>$2 \omega_1(f, (\sigma_2)^{1/2})$.</td>
    </tr>
    <tr>
      <td>3</td>
      <td>$(1 + (\sigma_2)^{1/2}/h) \omega_1(f, h)$.</td>
    </tr>
    <tr>
      <td>4</td>
      <td>$(1 + (\sigma_2)/h^2) \omega_1(f, h)$.</td>
    </tr>
    <tr>
      <td>5</td>
      <td>(Use ineq. 3 if $h&lt;(\sigma_2)^{1/2}$, or ineq. 4 otherwise.)</td>
    </tr>
    <tr>
      <td>6</td>
      <td>$\tilde\omega_1(f, (\sigma_2)^{1/2})$.</td>
    </tr>
  </tbody>
</table>

<p><em>Proof:</em> Inequality 1 follows from a result of Gonska and Meier (1985, theorem 3.1)<sup id="fnref:13"><a href="#fn:13" class="footnote" rel="footnote" role="doc-noteref">13</a></sup>. Inequality 2 follows from a result of Shisha and Mond (1968, theorem 1)<sup id="fnref:14"><a href="#fn:14" class="footnote" rel="footnote" role="doc-noteref">14</a></sup>; inequality 4 comes from another result in the same paper (see also Mamedov (1959)<sup id="fnref:15"><a href="#fn:15" class="footnote" rel="footnote" role="doc-noteref">15</a></sup>); inequality 3 follows from a result of Mond (1978)<sup id="fnref:16"><a href="#fn:16" class="footnote" rel="footnote" role="doc-noteref">16</a></sup>; inequality 5, a result of Păltănea (2004, corollary 1.2.2)<sup id="fnref:17"><a href="#fn:17" class="footnote" rel="footnote" role="doc-noteref">17</a></sup>; inequality 6, a result of Peetre (1969)<sup id="fnref:18"><a href="#fn:18" class="footnote" rel="footnote" role="doc-noteref">18</a></sup> (also mentioned in Gonska (1998/2023)<sup id="fnref:19"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>, which has an extensive discussion on error bounds for linear operators). □</p>

<p><strong>Remark 1:</strong> The moduli of continuity $\omega_1(f, \delta)$ and $\tilde\omega_1(f, \delta)$ offer concise ways to express different error bounds depending on how “regular” $f$ is.  Properties of these moduli are given in Sevy 1991<sup id="fnref:20"><a href="#fn:20" class="footnote" rel="footnote" role="doc-noteref">20</a></sup>, sec. 2.0.2; Gonska 1985<sup id="fnref:21"><a href="#fn:21" class="footnote" rel="footnote" role="doc-noteref">21</a></sup>. For example, let $f$ be continuous on a closed interval.  Then:</p>

<ul>
  <li>$\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)$ (Peetre 1969)<sup id="fnref:22"><a href="#fn:22" class="footnote" rel="footnote" role="doc-noteref">22</a></sup>.</li>
  <li>If $f$ is Hölder continuous with Hölder exponent $\alpha$ ($0\lt\alpha\le 1$) and Hölder constant $M$ or less, $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)\le M\delta^\alpha$.  Indeed, in this case, $f$ admits the continuous and concave modulus of continuity $\omega_1(\delta)=M\delta^\alpha$, where $\delta&gt;0$.</li>
  <li>If $f$ is Lipschitz continuous with Lipschitz constant $M$ or less, $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)\le M\delta$, given that Lipschitz-continuous functions are Hölder continuous with Hölder exponent 1.  The same bound holds true if $f$ instead has a continuous derivative with maximum absolute value $M$ or less, since in this case (by a result of Hardy and Littlewood) $f$ is Lipschitz continuous with Lipschitz constant $M$ or less.</li>
</ul>

<p>□</p>

<blockquote>
  <p><strong>Example:</strong> Let $f$ and $L$ be as in Lemma 1. If $f$ is Lipschitz continuous with Lipschitz constant $M$ or less, or has a continuous derivative with maximum absolute value $M$ or less, $\text{abs}(L(f)(\lambda)-f(\lambda))\le M (\sigma_2)^{1/2}$; this follows from the combination of Remark 1 and inequality 6 of Lemma 1.</p>
</blockquote>

<p><strong>Lemma 2</strong>. <em>Let $f(\lambda)$ be continuous on a closed interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind and reproduces all polynomials up to degree 1 (constants and linear functions).  Let $h&gt;0$ be a real number. Then:</em></p>

<table>
  <thead>
    <tr>
      <th>No.</th>
      <th>If $f$ …</th>
      <th>Then $\text{abs}(L(f)(\lambda)-f(\lambda))\le … $</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>1</td>
      <td>Has a continuous derivative.</td>
      <td>$((h+2)^2/(8h))\cdot \omega_1(f^{(1)}, h\cdot\sqrt{\sigma_2}) \cdot\sqrt{\sigma_2}$.</td>
    </tr>
    <tr>
      <td>2</td>
      <td>Has a continuous derivative.</td>
      <td>$\frac{1}{2}(\sigma_2)^{1/2} \tilde\omega_1(f^{(1)}, (\sigma_2)^{1/2})$.</td>
    </tr>
    <tr>
      <td>3</td>
      <td>Has a Hölder-continuous derivative with Hölder exponent $\alpha$ ($0\lt\alpha\le 1$) and Hölder constant $M$ or less.</td>
      <td>$\frac{M}{2}(\sigma_2)^{(1+\alpha)/2}$.</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Has a Lipschitz-continuous derivative with Lipschitz constant $M$ or less, or has a continuous second derivative with maximum absolute value $M$ or less.</td>
      <td>$\frac{M}{2} (\sigma_2)$.</td>
    </tr>
  </tbody>
</table>

<p><em>Proof:</em>  Inequality 1 is a special case of Theorem 2.19 (in conjunction with Remark 2.21) of Anastassiou (1985), with the interval $[a, b]$, $m=1$ (since the function is defined on all of $[a, b]$), $r=h$, and $x_0$ equal to $\lambda$.  Inequality 2 follows from a result of Gonska and Meier (1985, theorem 4.1)<sup id="fnref:13:1"><a href="#fn:13" class="footnote" rel="footnote" role="doc-noteref">13</a></sup>; see also Păltănea and Dimitriu (2016, remark 3)<sup id="fnref:23"><a href="#fn:23" class="footnote" rel="footnote" role="doc-noteref">23</a></sup>. Inequalities 3 and 4 follow from inequality 2 because of Remark 1. □</p>

<p><strong>Lemma 3</strong>. <em>Let $f(\lambda)$ have a continuous $k$-th derivative on a closed interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind.  Let $h&gt;0$ be a real number. Then $L(f)(\lambda) = L(Q_k(f,\lambda))(\lambda) + L(R_k(f,\lambda))(\lambda)$, where:</em></p>

\[\text{abs}(L(Q_k(f,\lambda)) - f(\lambda))\le \left(\sum_{i=0}^k \frac{\max(\text{abs}(f^{(i)})) \text{abs}(\sigma_i)}{i!}\right),\]

\[\text{abs}(L(R_k(f,\lambda)))\le\left(\frac{\tau_k}{k!}+\frac{\tau_{k+1}}{(k+1)!\cdot h}\right)\cdot\omega_1(f^{(k)}, h),\]

\[\text{and }\text{abs}(L(R_k(f,\lambda)))\le\max\left(\frac{\tau_k}{k!}, \frac{\tau_{k+1}}{(k+1)!\cdot 2h}\right)\cdot\tilde\omega_1(f^{(k)}, 2h),\]

\[\text{and }\text{abs}(L(R_k(f,\lambda)))\le\frac{\tau_k}{k!}\cdot\tilde\omega_1(f^{(k)}, \frac{\tau_{k+1}}{(k+1)\tau_k}),\]

<p><em>and where:</em></p>

<ul>
  <li>$Q_k(f,\lambda)=$ $\sum_{i=0}^k f^{(i)}(\lambda)\cdot(e_0-\lambda)^i/(i!)$ <em>is the degree-$k$</em> Taylor polynomial <em>of $f$ centered at $\lambda$.</em></li>
  <li>$R_k(f,\lambda)$ <em>is the</em> Taylor remainder <em>that results from subtracting $Q(f,\lambda)$ from $f$.</em></li>
</ul>

<p><em>Proof:</em>  The second to fourth bounds given relate to the Taylor remainder.  The second bound comes from Păltănea and Smuc (2019, Theorem 1)<sup id="fnref:24"><a href="#fn:24" class="footnote" rel="footnote" role="doc-noteref">24</a></sup>; the third bound comes from corollary 3.2 of Dimitriu (2010)<sup id="fnref:25"><a href="#fn:25" class="footnote" rel="footnote" role="doc-noteref">25</a></sup> and Brudnyĭ’s lemma; and the fourth bound follows from the second with $h=\tau_{k+1}/(2(k+1)\tau_k)$ and comes from Gonska et al. (2006)<sup id="fnref:26"><a href="#fn:26" class="footnote" rel="footnote" role="doc-noteref">26</a></sup>, where the closed interval assumed was the closed unit interval; see also Gonska (2007)<sup id="fnref:27"><a href="#fn:27" class="footnote" rel="footnote" role="doc-noteref">27</a></sup>, Piţul (2007)<sup id="fnref:11:1"><a href="#fn:11" class="footnote" rel="footnote" role="doc-noteref">11</a></sup>.  See also Anastassiou (1985, theorem 2.31)<sup id="fnref:28"><a href="#fn:28" class="footnote" rel="footnote" role="doc-noteref">28</a></sup>.<sup id="fnref:29"><a href="#fn:29" class="footnote" rel="footnote" role="doc-noteref">29</a></sup>□</p>

<p><strong>Lemma 4.</strong> <em>Let $k$ be zero or a positive integer. Let $f(\lambda)$—</em></p>

<ol>
  <li><em>have a Lipschitz-continuous $k$-th derivative on a closed interval, with Lipschitz constant $M$ or less, or</em></li>
  <li><em>have a continuous $(k+1)$-th derivative on that interval, with maximum absolute value $M$ or less,</em></li>
</ol>

<p><em>and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind.  Then</em> $\text{abs}(L(R_k(f,\lambda)))\le M \tau_{k+1}/((k+1)!)$, <em>where</em> $R_k(f,\lambda)$ <em>is as in Lemma 3.</em></p>

<p><em>Proof:</em>  Follows from the third bound for $L(R_k(f,\lambda))$ in Lemma 3 in the same manner as inequality 10 of Lemma 2, using Remark 1. □</p>

<p>The following two lemmas are more general, but not as easy to use.  In both, $\Vert L\Vert$ is the operator norm of $L$.</p>

<p><strong>Lemma 4A</strong> (special case of Theorem 3.4 in Gonska (1998/2023)<sup id="fnref:19:1"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>). <em>Let $f(\lambda)$ be continuous on a closed interval or a closed subset thereof, and let $L$ be a positive linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain.  Let $h&gt;0$ be a real number.  Then:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le\max(\Vert L\Vert ,L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_1(f,h)\]

\[+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda))\]

\[\le(\Vert L\Vert +L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_(f,h)+\text{abs}(L(e_0)(\lambda)-f(\lambda))\cdot\text{abs}(f(\lambda)),\]

<p><strong>Lemma 4B</strong> (special case of Theorem 4.7 in Gonska (1998/2023)<sup id="fnref:19:2"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>). <em>Let $f(\lambda)$ be continuous on a closed interval, and let $L$ be a positive linear operator that maps bounded functions on $f$’s domain to bounded functions on that domain.  Let $h&gt;0$ be a real number.  Then:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)+L(\text{ceil}((e_0-\lambda)/h-1))(\lambda))\cdot\omega_1(f,h)\]

\[+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\]

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)+L(\text{abs}(e_0-\lambda))(\lambda)/h)\cdot\omega_1(f,h)\]

\[+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)).\]

<p><em>The second inequality also works if $L$ maps from continuous functions instead of from bounded functions.</em></p>

<blockquote>
  <p><strong>Note:</strong> Unlike Lemma 4A, Lemma 4B doesn’t work for arbitrary closed subsets of $f$’s domain (see Remark 2.5 in Gonska (1998/2023)<sup id="fnref:19:3"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>.</p>
</blockquote>

<p>The following lemma adapts the previous lemmas to the setting of random variables.</p>

<p><strong>Lemma 5.</strong> <em>Let $f(\lambda)$ be continuous on a closed interval, and let $Y$ be a random variable taking only values in that interval.  Then Lemmas 1 through 4B apply as appropriate to $f$ meeting their conditions, with $L(f)=\mathbb{E}[f(Y)]$ and $\lambda =\mathbb{E}[Y]$.</em></p>

<p><em>Proof</em>: With these assumptions there is a positive linear operator $L(f) = \mathbb{E}[f(Y)]$ for $Y$ and $f$, according to Theorem 3.1.1 of Frantz (1984)<sup id="fnref:30"><a href="#fn:30" class="footnote" rel="footnote" role="doc-noteref">30</a></sup>, letting $x_o = \lambda$.  Then $L(e_0)$ = $\mathbb{E}[e_0(Y)]$ = $\mathbb{E}[1]$ = 1 regardless of $Y$, and  $L(e_1)$ = $\mathbb{E}[e_1(Y)]$ = $\mathbb{E}[Y]$ = $\lambda$, so $L$ reproduces all polynomials of degree up to 1. □</p>

<p><a id="Bounds_for_Remainder_of_Bernstein_Polynomials"></a></p>

<h3 id="bounds-for-remainder-of-bernstein-polynomials">Bounds for Remainder of Bernstein Polynomials</h3>

<p>The following results specialize the previous ones to the case of <a href="#Bernstein_Form_and_Bernstein_Polynomials"><strong>Bernstein polynomials</strong></a> $B_n$.  They apply to the Bernstein polynomial of the result of subtracting a Taylor polynomial from a function, and are useful when a linear operator contains $B_n(f)$ in its definition and reproduces all polynomials of degree $r$ or less.</p>

<p><strong>Lemma 6</strong>: <em>Let $k$ be zero or a positive integer.  Let $f(\lambda)$—</em></p>

<ol>
  <li><em>have a Lipschitz-continuous $k$-th derivative on the closed unit interval, with Lipschitz constant $M$ or less, or</em></li>
  <li><em>have a continuous $(k+1)$-th derivative on that interval, with maximum absolute value $M$ or less.</em></li>
</ol>

<p><em>Then the following bound holds true:</em> $\text{abs}(B_n(R_k(f, \lambda)) \le (M \mu_{k+1})/ ( ((k+1)!) n^{(k+1)/2})$ <em>for every integer $n\ge 2$ (and also for $n=1$ if $k$ is odd), where</em> $\mu_k$ <em>is as defined in Proposition 1.</em></p>

<p><em>Proof</em>: Follows from Lemma 4, with $L(f)=B_n(f)$, and from Proposition 1. □</p>

<p><strong>Corollary 1</strong>: <em>Let $f(\lambda)$, $k$, and $M$ be as in Lemma 6.  Then, for every $0\le\lambda\le 1$:</em></p>

<table>
  <thead>
    <tr>
      <th>If $k$ is:</th>
      <th>Then $\text{abs}(B_n(R_k(f, \lambda))) \le$ …</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0.</td>
      <td>$M(1/2)/n^{1/2}$ for every integer $n\ge 1$.</td>
    </tr>
    <tr>
      <td>1.</td>
      <td>$M(1/8)/n = 0.125M/n$ for every integer $n\ge 1$.</td>
    </tr>
    <tr>
      <td>2.</td>
      <td>$M(\sqrt{3}/48)/n^{3/2} &lt; 0.3609M/n^{3/2}$ for every integer $n\ge 2$.</td>
    </tr>
    <tr>
      <td>3.</td>
      <td>$M(1/128)/n^{2} = 0.0078125M/n^{2}$ for every integer $n\ge 1$.</td>
    </tr>
    <tr>
      <td>4.</td>
      <td>$M(\sqrt{5}/1280)/n^{5/2} &lt; 0.001747/n^{5/2}$ for every integer $n\ge 2$.</td>
    </tr>
    <tr>
      <td>5.</td>
      <td>$M(1/3072)/n^{3} &lt; 0.0003256/n^{3}$ for every integer $n\ge 1$.</td>
    </tr>
  </tbody>
</table>

<p><a id="Bounds_for_General_Linear_Operators"></a></p>

<h3 id="bounds-for-general-linear-operators">Bounds for General Linear Operators</h3>

<p>Roughly speaking, the <em>integral</em> of $f(\lambda)$ on the closed interval $[a,b]$ is the “area under the graph” of that function when the function is restricted to that interval.  If $f$ is continuous there, this is the value that—</p>

\[\frac{1}{n} \sum_{i=1}^n f\left(a+(b-a)(i-\frac{1}{2})/n\right),\tag{2A}\]

<p>approaches as $n$ gets larger and larger.  The integral of $f(\lambda)$ on $[a,b]$ is denoted $\int_a^b f(\lambda) d\lambda$.</p>

<p>Lemmas 7 and 8, which give error bounds for important classes of linear operators (not necessarily positive ones), rely on the so-called <em>Peano kernel theorem</em>, which was originally developed to assess the error in estimating the integral of a function from samples of it<sup id="fnref:31"><a href="#fn:31" class="footnote" rel="footnote" role="doc-noteref">31</a></sup> (for more on this theory, see Brass and Förster 1998<sup id="fnref:32"><a href="#fn:32" class="footnote" rel="footnote" role="doc-noteref">32</a></sup>; Waldron 1999<sup id="fnref:33"><a href="#fn:33" class="footnote" rel="footnote" role="doc-noteref">33</a></sup>).</p>

<p><strong>Lemma 7.</strong> <em>Let $k$ be zero or a positive integer, let $f(\lambda)$ have a continuous $(k+1)$-th derivative on the closed interval $[a, b]$, let $M$ be its maximum absolute value, and let $C$ and $c$ be real numbers such that $c\le f^{(k+1)}\le C$ over that interval. Let $L$ be a bounded linear operator that—</em></p>

<ul>
  <li><em>reproduces all polynomials of degree $k$ or less, and</em></li>
  <li><em>maps continuous functions (or, if $k=0$, bounded functions) on the interval $[a, b]$ to continuous functions on that interval.</em></li>
</ul>

<p><em>Then:</em></p>

\[\text{abs}(LF(f)(\lambda)) = \text{abs}(f(\lambda) - L(f)(\lambda))\]

\[\le \frac{C - c}{2} \frac{1}{k!}\int_a^b \text{abs}\left(LF((e_1-t)_+^k)(\lambda))\right) dt\tag{3}\]

\[= \frac{C - c}{2(k!)} \int_a^b \text{abs}\left((\lambda-t)_+^k-L((e_1-t)_+^k)(\lambda)\right) dt\tag{4}\]

\[\le \frac{M}{k!} \int_a^b \text{abs}\left((\lambda-t)_+^k-L((e_1-t)_+^k)(\lambda)\right) dt,\tag{5}\]

<p><em>where $LF(f) = f - L(f)$, and the notation</em> $(x)<em>+^k$ _is as follows. If $k\gt 0$, this equals $((x+\text{abs}(x))/2)^k$, or $\max(0, x)^k$, and $k$ is 0, this equals either 1 if $x\ge 0$ or 0 otherwise.</em></p>

<p>Formulas (3) and (4) are because, in this case, the operator $LF$ equals 0 on every polynomial of degree $k$ or less, so that $LF(e_i)=0$ whenever $0\le i\le k$, so that $LF$ satisfies theorem 3 of Gavrea and Ivan (2015)<sup id="fnref:34"><a href="#fn:34" class="footnote" rel="footnote" role="doc-noteref">34</a></sup>. Formula (5) is an easy consequence of (4); see also Brass and Förster (1998, theorem 5)<sup id="fnref:32:1"><a href="#fn:32" class="footnote" rel="footnote" role="doc-noteref">32</a></sup>.<sup id="fnref:35"><a href="#fn:35" class="footnote" rel="footnote" role="doc-noteref">35</a></sup></p>

<p><strong>Lemma 8</strong> (see Theorem 4 of Gavrea and Ivan (2015)<sup id="fnref:34:1"><a href="#fn:34" class="footnote" rel="footnote" role="doc-noteref">34</a></sup>). <em>With the assumptions in Lemma 7, if $LF$ is the difference of two positive linear operators $LA$ and $LB$, so that $LF(f)=LA(f)-LB(f)$ (or $L(f)=f-LA(f)+LB(f)$), and $LA$ and $LB$ both map continuous functions on that interval to functions of that kind, then:</em></p>

\[\text{abs}(L(f)(\lambda) - f(\lambda))\le \frac{C - c}{(k+1)!} \text{abs}(LA(e_{k+1})(\lambda)) \le\frac{2M}{(k+1)!} \text{abs}(LA(e_{k+1})(\lambda)).\]

<p><strong>Lemma 9</strong> (special case of Theorem 3.2 in Gonska (1998/2023)<sup id="fnref:19:4"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>). <em>Let $f(\lambda)$ be continuous on a closed interval or a closed subset thereof, and let $L$ be a bounded linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain.  Let $h&gt;0$ be a real number.  Then for each $\lambda$ in $f$’s domain:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((\Vert L\Vert+\alpha)/2, (\gamma(\beta(\lambda)-L(e_0)(\lambda))+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)\]

\[\cdot\tilde\omega_1(f,h)+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\]

<p><em>where $\alpha$ is the maximum of</em> $\text{abs}(L(e_0))$ <em>over $f$’s domain; $\beta(\lambda)$ is the maximum of $\text{abs}(L(g)(\lambda))$ over all continuous functions $g$ on $f$’s domain with a maximum absolute value of 1 or less; and $\gamma$ is the difference between the highest and lowest value of $\lambda$ in $f$’s domain.</em></p>

<p><strong>Lemma 10.</strong> <em>With the assumptions in Lemma 9, if $L$ reproduces constants, so that</em> $L(e_0)=1$, <em>the inequality in that lemma becomes:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((1+\Vert L\Vert)/2, (\gamma(\beta(\lambda)-1)+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)\cdot\tilde\omega_1(f,h).\]

<p><strong>Lemma 11</strong> (special case of Theorem 4.4 and Corollary 4.5 in Gonska (1998/2023)<sup id="fnref:19:5"><a href="#fn:19" class="footnote" rel="footnote" role="doc-noteref">19</a></sup>). <em>Let $f(\lambda)$ be continuous on a closed interval $[a, b]$, and let $L$ be a bounded linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain.  Let $h&gt;0$ be a real number.  Then for each $\lambda$ in $f$’s domain:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-\text{abs}(L(e_0)(\lambda)))\cdot(1+(b-a)/h)\]

\[+\text{abs}(L(e_0)(\lambda))+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)\]

\[+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\]

<p><em>where $\beta(\lambda)$ is as in Lemma 9.</em></p>

<p><strong>Lemma 12.</strong> <em>With the assumptions in Lemma 11, if $L$ reproduces constants, so that</em> $L(e_0)=1$, <em>the inequality in that lemma becomes:</em></p>

\[\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-1)\cdot(1+(b-a)/h)\]

\[+1+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)\]

<p><a id="Whitney_s_Inequality_on_Polynomial_Errors"></a></p>

<h3 id="whitneys-inequality-on-polynomial-errors">Whitney’s Inequality on Polynomial Errors</h3>

<p>The following inequality gives a bound on the “best possible” error that a polynomial of degree $n$ can achieve in approximating a function.</p>

<p>Let $n$ be zero or a positive integer, let $f(\lambda)$ be continuous on a closed interval $[a, b]$, and let $P$ be a polynomial of degree $n$ or less with the least maximum absolute difference between $f$ and the polynomial on that interval.  Then the error of $P$ in approximating $f$ is bounded as follows (see Babenko and Kryakin 2019<sup id="fnref:36"><a href="#fn:36" class="footnote" rel="footnote" role="doc-noteref">36</a></sup>):</p>

\[\Vert f-P\Vert _\infty\le W \cdot \omega_{n+1}(f,\frac{b-a}{n+1}),\]

<p>where—</p>

<ul>
  <li>$W$ is:
    <ul>
      <li>1 if $n\le 7$.</li>
      <li>$(2+\exp(-2)) (&lt; 2.13534)$ if $n\ge 8$.</li>
      <li>$3/4$ if $n=1$ and $f$ is convex (Singh Kaire and Prymak 2023/2025)<sup id="fnref:37"><a href="#fn:37" class="footnote" rel="footnote" role="doc-noteref">37</a></sup>.</li>
      <li>$1/2$ if $n=1$, $f$ is convex, and $a=-b$ (Singh Kaire and Prymak 2023/2025)<sup id="fnref:37:1"><a href="#fn:37" class="footnote" rel="footnote" role="doc-noteref">37</a></sup>.</li>
    </ul>
  </li>
  <li>$\omega_{n}(f, h)$ is the smallest modulus of continuity of $f$ of order $n$, with parameter $h$.</li>
</ul>

<p>Using properties of moduli of continuity (see Sevy 1991<sup id="fnref:20:1"><a href="#fn:20" class="footnote" rel="footnote" role="doc-noteref">20</a></sup>, sec. 2.0.2; Gonska 1985<sup id="fnref:21:1"><a href="#fn:21" class="footnote" rel="footnote" role="doc-noteref">21</a></sup>), if $f$ has a continuous $(n+1)$-th derivative on $[a, b]$:</p>

\[\Vert f-P\Vert _\infty\le W \cdot \left(\frac{b-a}{n+1}\right)^{n+1}\Vert f^{(n+1)}\Vert _\infty,\]

<p>and if $f$ has a continuous $n$-th derivative on that interval:</p>

\[\Vert f-P\Vert _\infty\le W \cdot \left(\frac{b-a}{n+1}\right)^n\omega_1(f^{(n)}, \frac{b-a}{n+1}).\]

<p><a id="Another_Inequality_on_Polynomial_Errors"></a></p>

<h3 id="another-inequality-on-polynomial-errors">Another Inequality on Polynomial Errors</h3>

<p>Like Whitney’s inequality, the following gives a bound on the “best possible” error between a polynomial and a function.</p>

<p>Let $n$ be zero or a positive integer, let $f(\lambda)$ have a continuous $(n+1)$-th derivative on the closed interval $[-1, 1]$,<sup id="fnref:38"><a href="#fn:38" class="footnote" rel="footnote" role="doc-noteref">38</a></sup> and let $P$ be a polynomial of degree $n$ or less with the least maximum absolute difference between $f$ and the polynomial on that interval. Then the error of $P$ in approximating $f$ is bounded as follows (Phillips 2003, theorem 2.4.6)<sup id="fnref:46"><a href="#fn:46" class="footnote" rel="footnote" role="doc-noteref">39</a></sup>:</p>

\[\Vert f-P\Vert _\infty\le\frac{1}{2^n}\frac{\Vert f^{(n+1)}\Vert _\infty}{(n+1)!}.\]

<p><a id="Lebesgue_Inequality_for_Certain_Linear_Operators"></a></p>

<h3 id="lebesgue-inequality-for-certain-linear-operators">Lebesgue Inequality for Certain Linear Operators</h3>

<p>Let $f(\lambda)$ be a continuous function on a closed interval.  For any sequence of linear operators $(L_n)$ that map continuous functions to polynomials and reproduce all polynomials up to degree $m(n)$ (which depends on $n$), the following error bound (also known as <em>Lebesgue’s lemma</em> or the <em>Lebesgue inequality</em>) holds true for each $n$:</p>

\[\text{abs}(L_n(f)(x) - f(x))\le(1+\Vert L_n\Vert )\cdot\max_t(\text{abs}(f(t)-P(t))),\]

<p>where $\Vert L_n\Vert$ is the operator norm of $L_n$, and $P$ is a polynomial of degree up to $m(n)$ with the least maximum absolute difference between $f$ and the polynomial (see also DeVore and Lorentz (1993)<sup id="fnref:39"><a href="#fn:39" class="footnote" rel="footnote" role="doc-noteref">40</a></sup>, Cheney (1996, chapter 6)<sup id="fnref:40"><a href="#fn:40" class="footnote" rel="footnote" role="doc-noteref">41</a></sup>).  But this error bound will generally be crude or trivial unless $L_n$ are nonpositive operators.  Indeed, the only positive linear operator $L$ that reproduces all polynomials up to degree 2 is the identity operator $L=f$.<sup id="fnref:41"><a href="#fn:41" class="footnote" rel="footnote" role="doc-noteref">42</a></sup></p>

<blockquote>
  <p><strong>Example:</strong> Let $f$ have a continuous third derivative on the closed unit interval.  Combining the previous inequality with the Whitney-type inequalities given earlier leads to the following error bound for linear operators $L$ that map continuous functions to polynomials and reproduce all polynomials up to degree 2:</p>

\[\text{abs}(L(f)(x) - f(x))\le(1+\Vert L\Vert )\cdot 1\cdot \left(\frac{1}{3}\right)^{3}\Vert f^{(3)}\Vert _\infty\]

\[= (1+\Vert L\Vert )\Vert f^{(3)}\Vert _\infty/27.\]
</blockquote>

<p><a id="Bounds_for_Certain_Nonlinear_Operators"></a></p>

<h3 id="bounds-for-certain-nonlinear-operators">Bounds for Certain Nonlinear Operators</h3>

<p>The following comes from a result in Bede and Gal (2010)<sup id="fnref:42"><a href="#fn:42" class="footnote" rel="footnote" role="doc-noteref">43</a></sup>; see also Bede et al. (2009)<sup id="fnref:43"><a href="#fn:43" class="footnote" rel="footnote" role="doc-noteref">44</a></sup>.</p>

<p>Let $f(\lambda)$ be continuous, bounded, and nonnegative on an interval.  Let $L$ be an operator that maps functions of that kind to functions of that kind and also has the following properties:</p>

<ol>
  <li>(Monotone.) For every pair of allowed functions $g$ and $h$, if $g\le h$, then $L(g)\le L(h)$.</li>
  <li>(Subadditive.) For every pair of allowed functions $g$ and $h$, $L(g+h)\le L(g)+L(h)$.</li>
  <li>(Positively homogeneous.) $xL(g)=L(xg)$ for every allowed function $g$ and every $x\ge 0$.</li>
</ol>

<p>If $L(e_0)=1$, then for every $h&gt;0$:</p>

\[\text{abs}(f(x)-L(f)(x))\le(1+L(\text{abs}(e_0-x))(x)/h)\cdot\omega_1(f, h),\]

<p>provided $L(\text{abs}(e_0-x))(x)$ (the “absolute moment” of $L$) exists (and is finite or infinite).</p>

<blockquote>
  <p><strong>Notes:</strong> An operator meeting conditions 2 and 3 is also called a <em>sublinear</em> operator.  Every linear operator is also sublinear. A linear operator is monotone if and only if it is positive.  For more on nonlinear operators, see Gal and Niculescu (2023)<sup id="fnref:44"><a href="#fn:44" class="footnote" rel="footnote" role="doc-noteref">45</a></sup>.</p>
</blockquote>

<p><a id="Example"></a></p>

<h2 id="example">Example</h2>

<p>This example shows how to find a linear operator’s bounds.</p>

<p>Let $L_n(f)$ be a linear operator inspired by <a href="https://peteroupc.github.io/bernsupp.html#A_Conjecture_on_Polynomial_Approximation"><strong>a conjecture I have</strong></a> on polynomial approximation.  It is described as follows:</p>

\[L_n(f)(\lambda) = \sum_{i=0}^n \left( W_{2n}\left(f\right)\left(\frac{k}{2n}\right) - W_n\left(f\right)\left(\frac{i}{n}\right)\right)\sigma_{n,k,i}\]

\[=\mathbb{E}\left[W_{2n}\left(f\right)\left(\frac{k}{2n}\right) - W_n\left(f\right)\left(\frac{X_k}{n}\right)\right],\]

<p>where:</p>

<ul>
  <li>$k = 2n\lambda$, where $0\le\lambda\le 1$.</li>
  <li>$W_n(f)$ is a linear operator that approaches $f$ as $n$ increases.<sup id="fnref:45"><a href="#fn:45" class="footnote" rel="footnote" role="doc-noteref">46</a></sup></li>
  <li>$X_k$ is a hypergeometric($2n$, $k$, $n$) random variable.</li>
  <li>$\sigma_{n,k,i}$ equals ${n\choose i}{n\choose {k-i}}/{2n \choose k}$ and is the probability that $X_k$ equals $i$.</li>
  <li>$\mathbb{E}[Y]$ is the expected value (or mean or “long-run average”) of the random variable $Y$.</li>
</ul>

<p>$L_n$ and $W_n$ are generally nonpositive operators.  As an example, take $W_n=2f-B_n(f)$.  Then $B_n(W_n(f))$ is a linear operator that is the iterated Boolean sum of degree-$n$ Bernstein polynomials, with one iteration; see Güntürk and Li (2021a, Theorem 5)<sup id="fnref:46:1"><a href="#fn:46" class="footnote" rel="footnote" role="doc-noteref">39</a></sup>.  That paper, among others (for example, Micchelli 1973<sup id="fnref:47"><a href="#fn:47" class="footnote" rel="footnote" role="doc-noteref">47</a></sup>), showed that this operator approaches $f$ at the rate $O(1/n^{3/2})$ if $f$ has a continuous third derivative. (“$O(1/n^{3/2})$” means the error is no greater than a constant times $1/n^{3/2}$ for all values of $n$.)</p>

<p>With this choice of $W_n$, $L_n$ becomes:</p>

\[L_n(f)(\lambda) = \sum_{i=0}^n\left((2f\left(\frac{k}{2n}\right) - B_{2n}(f)\left(\frac{k}{2n}\right)) - (2f\left(\frac{i}{n}\right) - B_n(f)\left(\frac{i}{n}\right))\right) \sigma_{n,k,i}\]

\[= \mathbb{E}\left[(2f\left(\frac{k}{2n}\right) - B_{2n}(f)\left(\frac{k}{2n}\right)) - (2f\left(\frac{X_k}{n}\right) - B_n(f)\left(\frac{X_k}{n}\right))\right]\]

\[= \sum_{i=0}^n\left((2f\left(\frac{k}{2n}\right) + B_{n}(f)\left(\frac{i}{n}\right))\right)\sigma_{n,k,i} - \sum_{i=0}^n \left((2f\left(\frac{i}{n}\right) + B_{2n}(f)\left(\frac{k}{n}\right))\right) \sigma_{n,k,i}\]

\[= LA_n(f)(\lambda) - LB_n(f)(\lambda).\]

<p>Here, $LA_n$ and $LB_n$ are positive linear operators, making it easier to assess their approximation properties.</p>

<p>It will be shown that, if $f$ has a continuous third derivative, the rate of $L_n$ towards zero is $O(M/n^{3/2})$, where $M$ is the maximum absolute value of $f$ and its derivatives up to the third derivative.  The proof of this relies on exact expressions of $L_n$’s <a href="#Moments_of_Linear_Operators"><strong>“raw moments” and “central moments”</strong></a>, and those for the combined operator $(LA_n+LB_n)$.</p>

<p>The following are some of these values and those for related operators:</p>

<ul>
  <li>$L_n(e_0)(x) = L_n((e_1-x)^0)(x) = 0$.</li>
  <li>$L_n(e_1)(x) = L_n((e_1-x)^1)(x) = 0$.</li>
  <li>$L_n(e_2)(x) = L_n((e_1-x)^2)(x)$ = $3x(x - 1)/(2n(2n-1))$ = $O(1/n^2)$.</li>
  <li>$L_n(e_3)(x)$ = $n^3 x^2(2nx - 4x + 3)/(2n - 1)$.</li>
  <li>$LA_n((e_1-x)^2)(x)$ = $-x(3n - 2)\cdot(x - 1)/(n(2n-1))$ = $O(1/n)$.</li>
  <li>$LB_n((e_1-x)^2)(x)$ = $-x(6n - 1)\cdot(x - 1)/(2n(2n-1))$ = $O(1/n)$.</li>
  <li>$(LA_n+LB_n)((e_1-x)^2)(x)$ = $LA_n(\text{abs}(e_1-x)^2)(x) + LB_n(\text{abs}(e_1-x)^2)(x)$ = $-x(12n - 5)\cdot(x - 1)/(2n(2n - 1)) = O(1/n)$.</li>
</ul>

<p>To find values like those just listed, it is useful to calculate raw moments (Wang et al. 2023)<sup id="fnref:48"><a href="#fn:48" class="footnote" rel="footnote" role="doc-noteref">48</a></sup> and central moments (Weisstein)<sup id="fnref:49"><a href="#fn:49" class="footnote" rel="footnote" role="doc-noteref">49</a></sup> of hypergeometric random variables (such as $X_k$).  Indeed, if $g(y)=W_{2n}(e_r;k/(2n))-W_n(e_r;y)$ is a polynomial in $y$ of degree $r$ or less, then $L_n(e_r)$ can be found using a Taylor expansion, namely as—</p>

\[L_n(e_r) = \sum_{i=0}^r \mathbb{E}[(X_k/n-\mathbb{E}[X_k/n])^i]\frac{g^{(i)}(\mathbb{E}[X_k/n])}{i!}\]

\[= \sum_{i=0}^r \frac{\mathbb{E}[(X_k-\mathbb{E}[X_k])^i]}{n^i}\frac{g^{(i)}(k/(2n))}{i!},\]

<p>where the derivatives are taken with respect to $y$, and where $\mathbb{E}[(X_k-\mathbb{E}[X_k])^i]$ is the $i$-th central moment of $X_k$.</p>

<p>In the following, the notation $\Vert f\Vert$ means $\max_{0\le\lambda\le 1}(\text{abs}(f(\lambda)))$.</p>

<p>The first step is to <a href="#Taylor_Expansion_of_Linear_Operators"><strong>find the Taylor expansion</strong></a> of $L_n(f)(\lambda)$. Given that $L_n((e_1-x)^0)(x)$ = $L_n((e_1-x)^1)(x)$ = 0, this becomes:</p>

\[L_n(f)(\lambda) = L_n(R_3(f, \lambda)) + \sum_{i=2}^3 L_n((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!},\]

\[\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert + \Vert L_n((e_1-\lambda)^2)\Vert  \Vert f^{(2)}\Vert /2\]

\[+ \Vert L_n((e_1-\lambda)^3)\Vert  \Vert f^{(3)}\Vert /6.\]

<p>The function $\text{abs}(L_n((e_1-x)^3)(x))$ has its maximum at $x=1/2-\sqrt{3}/6$; and $\text{abs}(L_n((e_1-x)^2)(x))$ has its maximum at $x=1/2$, so:</p>

\[\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert  + \text{abs}(\frac{3\lambda(\lambda - 1)}{2n(2n-1)})\Vert f^{(2)}\Vert /2\]

\[+ \Vert L_n((e_1-\lambda)^3)\Vert  \Vert f^{(3)}\Vert /6\]

\[\le \Vert L_n(R_3(f, \lambda))\Vert  + \frac{3}{8n(2n-1)}\Vert f^{(2)}\Vert /2\]

\[+ \frac{\sqrt{3} (6 n - 5)}{24 n^{2} (2 n - 1)}\Vert f^{(3)}\Vert /6.\]

<p>Meanwhile the remainder is estimated as follows, using the proof of corollary 2.3 of Gonska et al. (2006)<sup id="fnref:3:1"><a href="#fn:3" class="footnote" rel="footnote" role="doc-noteref">3</a></sup>:</p>

\[\Vert L_n(R(f, \lambda))\Vert \le \frac{1}{6} \Vert f^{(3)}\Vert  \Vert (LA_n+LB_n)(\text{abs}(e_1-\lambda)^3)\Vert .\]

<p>In turn, using Schwarz’s inequality (see proof of the same paper’s corollary 2.1):</p>

\[\Vert (LA_n+LB_n)(\text{abs}(e_1-\lambda)^3)\Vert \le (\Vert (LA_n+LB_n)((e_1-\lambda)^4)\Vert )^{1/2}\]

\[\times (\Vert (LA_n+LB_n)((e_1-\lambda)^2)\Vert )^{1/2} \le \frac{3\sqrt{3}}{8n^{3/2}}.\]

<p>(The expression in the middle takes its maximum at $\lambda = 1/2$; the right-hand side is an upper bound of that expression for all positive integers $n$.) Altogether:</p>

\[\Vert L_n(f)\Vert  \le \frac{3}{8n(2n-1)}\frac{1}{2}\Vert f^{(2)}\Vert\]

\[+ \left(\frac{3\sqrt{3}}{8n^{3/2}} + \frac{\sqrt{3} (6 n - 5)}{24 n^{2} (2 n - 1)}\right)\frac{1}{6}\Vert f^{(3)}\Vert  = LC_n(f)\]

\[\le 0.1875 \frac{\Vert f^{(2)}\Vert }{n^{3/2}} + \frac{5\sqrt{3}}{72} \frac{\Vert f^{(3)}\Vert }{n^{3/2}} \le \frac{0.3078 M}{n^{3/2}} = O(1/n^{3/2}).\]

<p>If $n\ge 2$ is an integer, $LC_n(f)\le 0.2165 M/n^{3/2}$.</p>

<p><a id="Example_An_Interesting_Linear_Operator"></a></p>

<h2 id="example-an-interesting-linear-operator">Example: An Interesting Linear Operator</h2>

<p>For a continuous function $f$ on the closed unit interval and for nonnegative integers $m$ and $n$, let $H_{n,m}$ be a linear operator as follows:</p>

\[H_{n,m}(f)=B_n(f) + \text{Lag}_m(f) - B_n(\text{Lag}_m(f)),\]

<p>where $B_n$ is the degree-$n$ Bernstein polynomial and $\text{Lag}_m$ is the polynomial of degree up to $m$ that equals $f$ at “$m+1$ distinct points on” the closed unit interval.  This operator was mentioned in Remark 2 of Gavrea and Ivan (2018)<sup id="fnref:50"><a href="#fn:50" class="footnote" rel="footnote" role="doc-noteref">50</a></sup>, but appears not to have been studied elsewhere.</p>

<p>It is known that $Lag_m$ is a linear operator and reproduces all polynomials of degree $m$ or less, so that $Lag_m(e_i) = e_i$ whenever $0\le i\le m$ is an integer. Thus, if $f$ is such a polynomial, $B_n(f)=B_n(Lag_m(f))$ and therefore $H_{n,m}(f)$ = $Lag_m(f)=f$, and therefore $H_{n,m}(e_i)=e_i$ whenever $0\le i\le m$ is an integer.</p>

<p>(The foregoing sentence would remain true if $B_n$ were replaced with any other operator mapping to and from the same functions.)</p>

<p>Because $H_{n,m}$ is linear and reproduces all polynomials up to degree $m$, the following holds if $f$ has a continuous $m$-th derivative:</p>

\[H_{n,m}(f)(\lambda) - f(\lambda) = H_{n,m}(R_m(f, \lambda))(\lambda)\]

\[=B_n(R_m(f,\lambda)) + \text{Lag}_m(R_m(f,\lambda)) - B_n(\text{Lag}_m(R_m(f,\lambda))).\]

<p>With the help of Lemma 6, the following holds if $n$ is also 2 or greater:</p>

\[\Vert H_{n,m}(f)(\lambda)\Vert  \le \frac{\Vert f^{(m)}\Vert  \mu_{r}}{ (r!) n^{r/2}} + \Vert \text{Lag}_m(R_m(f,\lambda)) - B_n(\text{Lag}_m(R_m(f,\lambda)))\Vert ,\]

<p>where $\mu_r$ is as in Proposition 1 and the notation $\Vert f\Vert$ means $\max_{0\le\lambda\le 1}(\text{abs}(f(\lambda)))$.</p>

<p>Alternatively, write:</p>

\[H_{n,m}(f) - f=B_n(f) + \text{Lag}_m(f) - B_n(\text{Lag}_m(f)) - f\]

\[=(B_n(f) - f) + (\text{Lag}_m(f) - B_n(\text{Lag}_m(f))),\]

\[\text{abs}((H_{n,m}(f) - f)(\lambda))\le\text{abs}(B_n(f) - f) + \text{abs}(B_n(\text{Lag}_m(f)) - \text{Lag}_m(f)),\]

<p>so now there are two error bounds to find: one for $f$ and the other for $\text{Lag}_m(f)$.  And, if $f$ has a continuous second derivative, both have the same form:</p>

\[B_n(g)\le M_2(g)/(8n),\]

<p>where $M_i(g)$ is the maximum absolute value of $g$’s $i$-th derivative. (This follows from Lorentz (1963)<sup id="fnref:51"><a href="#fn:51" class="footnote" rel="footnote" role="doc-noteref">51</a></sup> and the well-known fact that $M_2$ is an upper bound of $g$’s first derivative’s smallest Lipschitz constant.) Thus what is left is to estimate the second derivative of $\text{Lag}_m(f)$.  Given that that function is a polynomial of degree $m$ or less, this can be estimated as:</p>

\[\text{abs}(\text{Lag}_m(f)^{(2)}(\lambda))\le \Vert \text{Lag}_m\Vert  M_0(f)\cdot \max(1,m)^2,\]

<p>where $\Vert Lag_m\Vert$ is the operator norm of $Lag_m$, also known as its <em>Lebesgue constant</em>, which will vary depending on the points on the closed unit interval where the polynomial meets (interpolates) $f$.  The inequality just shown relies on Bernstein’s inequality for the derivatives of polynomials (Weisstein)<sup id="fnref:52"><a href="#fn:52" class="footnote" rel="footnote" role="doc-noteref">52</a></sup>.</p>

<p>Altogether, if $f$ has a continuous second derivative and $m$ is fixed:</p>

\[\text{abs}((H_{n,m}(f) - f)(\lambda))\le \frac{M_2(f)}{8n} + \frac{\Vert \text{Lag}_m\Vert  M_0(f)\cdot \max(1,m)^2}{8n}.\]

<hr />

<blockquote>

  <p><strong>Example:</strong> If $m$ is 3, and the polynomial generated by $Lag_m$ interpolates $f$ at the points 0, 1/3, 2/3, and 1, the inequality just shown becomes:</p>

\[\text{abs}((H_{n,3}(f) - f)(\lambda))\le \frac{M_2(f)}{8n} + \frac{1.64\cdot M_0(f)\cdot 9}{8n},\]

  <p>using an upper bound for $\Vert Lag_3\Vert$.</p>
</blockquote>

<p><a id="Probabilistic_Interpretations_of_Linear_Operators"></a></p>

<h2 id="probabilistic-interpretations-of-linear-operators">Probabilistic Interpretations of Linear Operators</h2>

<p>The Bernstein polynomials featured in a proof in 1912 of the result that any continuous function on a closed interval can be approximated as well as desired by polynomials (Bernstein 1912)<sup id="fnref:53"><a href="#fn:53" class="footnote" rel="footnote" role="doc-noteref">53</a></sup>. That proof used probability theory. In a series of papers, Adell and De la Cal use probability theory to interpret a number of linear operators in addition to those polynomials (Adell and De la Cal 1996<sup id="fnref:54"><a href="#fn:54" class="footnote" rel="footnote" role="doc-noteref">54</a></sup>, 1995<sup id="fnref:55"><a href="#fn:55" class="footnote" rel="footnote" role="doc-noteref">55</a></sup>).</p>

<p><a id="License"></a></p>

<h2 id="license">License</h2>

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<a id="Notes"></a></p>

<h2 id="notes">Notes</h2>

<div class="footnotes" role="doc-endnotes">
  <ol>
    <li id="fn:1">
      <p>A better term for positive operators is probably nonnegativity-preserving operators. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:2">
      <p><em>n</em>! = 1*2*3*…*<em>n</em> is also known as <em>n</em> factorial; in this document, (0!) = 1.<br /><em>Summation notation</em>, involving the Greek capital sigma (Σ), is a way to write the sum of one or more terms of similar form. For example, $\sum_{k=0}^n g(k)$ means $g(0)+g(1)+…+g(n)$, and $\sum_{k\ge 0} g(k)$ means $g(0)+g(1)+…$. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:3">
      <p>Gonska, Heiner, Paula Piƫul, and Ioan Raşa. “On differences of positive linear operators.” Carpathian Journal of Mathematics (2006): 65-78. <a href="#fnref:3" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:3:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:4">
      <p>Skorski, Maciej. “Handy formulas for binomial moments.” <em>Modern Stochastics: Theory and Applications</em> 12.1 (2024): 27-41. <a href="#fnref:4" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:5">
      <p>It is also possible to bound the “absolute moment” as $M_{n,r}(p)\le C(r)(\max(1/n, (p(1-p)/n)^{1/2})^r$ or $M_{n,r}(p)\le D(r)(1/n + (p(1-p)/n)^{1/2})^r$ (G.G. Lorentz, “The degree of approximation by polynomials with positive coefficients”, 1966), but the constants $C(r)$ and $D(r)$ seem to be higher (and less favorable) than the $E(r)$ in $M_{n,r}(p)\le E(r)/n^{r/2}$. <a href="#fnref:5" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:6">
      <p>Adell, J.A., Cárdenas-Morales, D., “<a href="https://www.sciencedirect.com/science/article/pii/S0021904518300376"><strong>Quantitative generalized Voronovskaja’s formulae for Bernstein polynomials</strong></a>”, Journal of Approximation Theory 231, July 2018. <a href="https://doi.org/10.1016/j.jat.2018.04.007"><strong>https://doi.org/10.1016/j.jat.2018.04.007</strong></a> <a href="#fnref:6" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:7">
      <p>Molteni, Giuseppe. “Explicit bounds for even moments of Bernstein’s polynomials.” Journal of Approximation Theory 273 (2022): 105658. <a href="#fnref:7" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:8">
      <p>Cheng, F., “On the rate of convergence of Bernstein polynomials of functions of bounded variation”, Journal of Approximation Theory 39 (1983). <a href="#fnref:8" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:9">
      <p>Weisstein, Eric W. “Schwarz’s Inequality.” From MathWorld–A Wolfram Resource. <a href="https://mathworld.wolfram.com/SchwarzsInequality.html"><strong>https://mathworld.wolfram.com/SchwarzsInequality.html</strong></a> <a href="#fnref:9" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:10">
      <p>R. Bojanić, O. Shisha, “Degree of $L^1$ approximation to integrable functions by modified Bernstein polynomials”, Journal of Approximation Theory 13, 66–72 (1975). <a href="#fnref:10" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:11">
      <p>Piţul, P., “Evaluation of the Approximation Order by Positive Linear Operators”, dissertation, Universität Duisberg-Essen, 2007. <a href="#fnref:11" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:11:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:12">
      <p>I suspect that, whenever $L$ is a bounded linear operator that maps continuous functions on a closed interval to functions of that kind, $L$ can be written as a difference between two positive linear operators.  But I have not seen a proof of that statement; Acu et al. (“<a href="https://doi.org/10.1007/s11253-011-0548-2"><strong>Grüss-type and Ostrowski-type inequalities in approximation theory</strong></a>”, Ukr Math J 63, 843–864, 2011) give a similar statement but without proof. <a href="#fnref:12" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:13">
      <p>Gonska, H.H., Meier, J., “On approximation by Bernstein-type operators: best constants”, Studia Sci. Math. Hungar. 22, 1987. <a href="#fnref:13" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:13:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:14">
      <p>Shisha, O., Mond. B, “The degree of convergence of linear positive operators”, 1968. <a href="#fnref:14" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:15">
      <p>R. G. Mamedov, “On the order of approximation of functions by linear positive operators” (Russian), Dokl. Akad. Nauk SSSR 128 (1959). <a href="#fnref:15" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:16">
      <p>Mond, B., “On the degree of approximation by linear positive operators”, <em>Journal of Approximation Theory</em> 18 (1976). <a href="#fnref:16" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:17">
      <p>Păltănea, R., <em>Approximation Theory Using Positive Linear Operators</em>, Birkhäuser, 2004. <a href="https://doi.org/10.1007/978-1-4612-2058-9"><strong>https://doi.org/10.1007/978-1-4612-2058-9</strong></a> <a href="#fnref:17" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:18">
      <p>Peetre, J., “On the connection between the theory of interpolation spaces and approximation theory”, in <em>Approximation Theory</em>, 1969. <a href="#fnref:18" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:19">
      <p>Gonska, Heiner. “The rate of convergence of bounded linear processes on spaces of continuous functions.” Journal of Numerical Analysis and Approximation Theory 52.2 (2023): 182-232. <a href="https://doi.org/10.33993/jnaat522-1326"><strong>https://doi.org/10.33993/jnaat522-1326</strong></a> <a href="#fnref:19" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:19:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a> <a href="#fnref:19:2" class="reversefootnote" role="doc-backlink">&#8617;<sup>3</sup></a> <a href="#fnref:19:3" class="reversefootnote" role="doc-backlink">&#8617;<sup>4</sup></a> <a href="#fnref:19:4" class="reversefootnote" role="doc-backlink">&#8617;<sup>5</sup></a> <a href="#fnref:19:5" class="reversefootnote" role="doc-backlink">&#8617;<sup>6</sup></a></p>
    </li>
    <li id="fn:20">
      <p>Sevy, J., “Acceleration of convergence of sequences of simultaneous approximants”, dissertation, Drexel University, 1991. <a href="https://doi.org/10.17918/00010296"><strong>https://doi.org/10.17918/00010296</strong></a> <a href="#fnref:20" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:20:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:21">
      <p>H. H. Gonska, <em>Quantitative Approximation in C(X)</em>, Habilitationschrift, Universität Duisburg, 1985. <a href="#fnref:21" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:21:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:22">
      <p>Peetre, J., “Exact interpolation theorems for Lipschitz-continuous functions”, Ricerche Mat. 18 (1969). <a href="#fnref:22" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:23">
      <p>Păltănea, R, Dimitriu, M.T., “On some second order moduli of smoothness.” General Mathematics 24 (2016) <a href="#fnref:23" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:24">
      <p>Păltănea, R., Smuc, M. “Sharp Estimates of Asymptotic Error of Approximation by General Positive Linear Operators in Terms of the First and the Second Moduli of Continuity”, <em>Results in Mathematics</em> 74, 70 (2019). <a href="https://doi.org/10.1007/s00025-019-0997-8"><strong>https://doi.org/10.1007/s00025-019-0997-8</strong></a> <a href="#fnref:24" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:25">
      <p>Dimitriu, M.T., “<a href="https://www.jstor.org/stable/43964559"><strong>Estimates with optimal constants using Peetre’s K-functionals</strong></a>”, <em>Carpathian Journal of Mathematics</em> 26 (2010). <a href="#fnref:25" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:26">
      <p>Gonska, Heiner, Paula Piţul, and Ioan Raşa. “On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators”. In <em>Proceedings of the International Conference on Numerical Analysis and Approximation Theory</em>, Cluj-Napoca. Romania, July 2006. <a href="#fnref:26" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:27">
      <p>Gonska, Heiner. “On the degree of approximation in Voronovskaja’s theorem”, Studia Univ. Babeş-Bolyai, Math., September 2007. <a href="#fnref:27" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:28">
      <p>Anastassiou, George A. “<a href="https://www.sciencedirect.com/science/article/pii/0021904585900498"><strong>A study of positive linear operators by the method of moments, one-dimensional case</strong></a>.” Journal of Approximation Theory 45.3 (1985): 247-270. <a href="#fnref:28" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:29">
      <p>The paper Cichoń et al., “<a href="https://doi.org/10.1137/1.9781611973037.11"><strong>On delta-method of moments and probabilistic sums</strong></a>”, ANALCO 2013, has very similar results, but they assume the function $f$ has a $k$-th derivative defined on an <em>open</em> interval (say, $0\lt\lambda\lt 1$), rather than a <em>closed</em> one, making those results harder to use if $Y$ is a random variable that can take a value equal to either endpoint of the interval (in this example, 0 or 1). <a href="#fnref:29" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:30">
      <p>Frantz, Deborah A. <a href="https://preserve.lehigh.edu/lehigh-scholarship/graduate-publications-theses-dissertations/theses-dissertations/summability"><strong>Summability methods, probability distributions, and associated positive linear operators</strong></a>. Lehigh University, 1984. <a href="#fnref:30" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:31">
      <p>This kind of estimation is called <em>quadrature</em> or <em>numerical integration</em>, and methods for such estimation, such as the one given in (2A), are called <em>quadrature rules</em>. <a href="#fnref:31" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:32">
      <p>Brass, H., Förster, KJ. (1998). On the Application of the Peano Representation of Linear Functionals in Numerical Analysis. In: Milovanović, G.V. (eds) Recent Progress in Inequalities. Mathematics and Its Applications, vol 430. Springer, Dordrecht. <a href="https://doi.org/10.1007/978-94-015-9086-0_10"><strong>https://doi.org/10.1007/978-94-015-9086-0_10</strong></a> <a href="#fnref:32" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:32:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:33">
      <p>Waldron, Shayne. “Refinements of the Peano kernel theorem.” Numerical functional analysis and optimization 20.1-2 (1999): 147-161. <a href="https://doi.org/10.1080/01630569908816885"><strong>https://doi.org/10.1080/01630569908816885</strong></a> <a href="#fnref:33" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:34">
      <p>Gavrea, I., Ivan, M., “A sharp estimate for the Peano error representation”, <em>Applied Mathematics and Computation</em> 252 (2015). <a href="https://doi.org/10.1016/j.amc.2014.12.017"><strong>https://doi.org/10.1016/j.amc.2014.12.017</strong></a> <a href="#fnref:34" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:34:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:35">
      <p>Note that for formulas (3) to (5), $(e_1-t)_+^0$ is discontinuous and so is not accepted by $LF$ and $L$ if they map from only continuous functions; thus the results in this section suppose both operators map from bounded functions for $k=0$.  Brass and Förster 1998 adequately provides for the case $k=0$, but not Gavrea and Ivan 2015, unfortunately. <a href="#fnref:35" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:36">
      <p>Babenko, Alexander G., and Yuriy V. Kryakin. “Special difference operators and the constants in the classical Jackson-type theorems.” Topics in Classical and Modern Analysis: In Memory of Yingkang Hu. Cham: Springer International Publishing, 2019. 35-46. <a href="#fnref:36" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:37">
      <p>Jaskaran Singh Kaire and Andriy Prymak, “<a href="https://arxiv.org/abs/2311.00912"><strong>Whitney-type estimates for convex functions</strong></a>”, arXiv:2311.00912 (2023). <a href="#fnref:37" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:37:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:38">
      <p>It would be interesting to find a version of this inequality that works for any closed interval $[a, b]$. <a href="#fnref:38" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:46">
      <p>Güntürk, C. Sinan, and Weilin Li. “<a href="https://arxiv.org/pdf/2112.09183"><strong>Approximation with one-bit polynomials in Bernstein form</strong></a>”, arXiv:2112.09183 (2021); Constr Approx 57, 601–630 (2023). <a href="https://doi.org/10.1007/s00365-022-09608-y"><strong>https://doi.org/10.1007/s00365-022-09608-y</strong></a> <a href="#fnref:46" class="reversefootnote" role="doc-backlink">&#8617;</a> <a href="#fnref:46:1" class="reversefootnote" role="doc-backlink">&#8617;<sup>2</sup></a></p>
    </li>
    <li id="fn:39">
      <p>R.A. DeVore and G.G. Lorentz, <em>Constructive Approximation</em>, 1993. <a href="https://link.springer.com/book/9783540506270"><strong>https://link.springer.com/book/9783540506270</strong></a> <a href="#fnref:39" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:40">
      <p>E. W. Cheney, <em>Introduction to Approximation Theory</em>, 1998. <a href="#fnref:40" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:41">
      <p>Guessab, A., Nouisser, O. &amp; Schmeisser, G. Enhancement of the algebraic precision of a linear operator and consequences under positivity. <em>Positivity</em> 13, 693–707 (2009). <a href="https://doi.org/10.1007/s11117-008-2253-4"><strong>https://doi.org/10.1007/s11117-008-2253-4</strong></a>. However, Gavrea and Ivan (“<a href="https://doi.org/10.1016/j.jat.2017.12.001"><strong>A note on the fixed points of positive linear operators</strong></a>”, <em>Journal of Approximation Theory</em> (227), 2018) pointed out that there are positive linear operators besides the identity that reproduce all polynomials of the form $x^i$ where $i&gt;0$. <a href="#fnref:41" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:42">
      <p>Bede, Barnabás, and Sorin G. Gal. “Approximation by Nonlinear Bernstein and Favard-Szász-Mirakjan Operators of Max-Product Kind.” Journal of Concrete &amp; Applicable Mathematics 8.1 (2010). <a href="#fnref:42" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:43">
      <p>Bede, Barnabás, Coroianu, Lucian, Gal, Sorin G., Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind, International Journal of Mathematics and Mathematical Sciences, 2009, 590589, 26 pages, 2009. <a href="https://doi.org/10.1155/2009/590589"><strong>https://doi.org/10.1155/2009/590589</strong></a> <a href="#fnref:43" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:44">
      <p>Gal, Sorin G., and Constantin P. Niculescu. “<a href="https://arxiv.org/abs/2206.14102v1"><strong>Korovkin-type theorems for weakly nonlinear and monotone operators</strong></a>”, arXiv:2206.14102v1 [math.FA], also in <em>Mediterranean Journal of Mathematics</em> 20.2 (2023): 56. <a href="https://doi.org/10.1007/s00009-023-02271-y"><strong>https://doi.org/10.1007/s00009-023-02271-y</strong></a> <a href="#fnref:44" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:45">
      <p>$W_n$ can, in principle, be nonlinear instead, but this would require a totally different approach to finding the approximation error, and $L_n$ would then be nonlinear in general. <a href="#fnref:45" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:47">
      <p>Micchelli, Charles. “<a href="https://www.sciencedirect.com/science/article/pii/0021904573900282"><strong>The saturation class and iterates of the Bernstein polynomials</strong></a>”, Journal of Approximation Theory 8, no. 1 (1973): 1-18. <a href="#fnref:47" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:48">
      <p>Wang, Y.Q., Zhang, Y.Y, Liu, J.L., “Expectation identity of the hypergeometric distribution and its application in the calculations of high-order origin moments”, Communications in Statistics–Theory and Methods 52(17), 2023. <a href="https://doi.org/10.1080/03610926.2021.2024235"><strong>https://doi.org/10.1080/03610926.2021.2024235</strong></a> <a href="#fnref:48" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:49">
      <p>Weisstein, Eric W. “Central Moment.” From MathWorld–A Wolfram Resource. <a href="https://mathworld.wolfram.com/CentralMoment.html"><strong>https://mathworld.wolfram.com/CentralMoment.html</strong></a> <a href="#fnref:49" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:50">
      <p>Ioan Gavrea, Mircea Ivan, “A note on the fixed points of positive linear operators”, Journal of Approximation Theory (227), 2018, <a href="https://doi.org/10.1016/j.jat.2017.12.001"><strong>https://doi.org/10.1016/j.jat.2017.12.001.</strong></a>. <a href="#fnref:50" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:51">
      <p>G.G. Lorentz, “Inequalities and saturation classes for Bernstein polynomials”, 1963. <a href="#fnref:51" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:52">
      <p>Weisstein, Eric W. “Bernstein’s Inequality.” From MathWorld–A Wolfram Resource. <a href="https://mathworld.wolfram.com/BernsteinsInequality.html"><strong>https://mathworld.wolfram.com/BernsteinsInequality.html</strong></a> <a href="#fnref:52" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:53">
      <p>S.N. Bernstein, “Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités”, Comm. Kharkov Math. Soc. 13, 1-2, 1912. <a href="#fnref:53" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:54">
      <p>Adell, J. A., and J. De la Cal. “Bernstein-type operators diminish the φ-variation.” Constructive Approximation 12.4 (1996): 489-507. <a href="https://doi.org/10.1007/BF02437505"><strong>https://doi.org/10.1007/BF02437505</strong></a> <a href="#fnref:54" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
    <li id="fn:55">
      <p>Adell, J. A., and J. De la Cal. “Bernstein-Durrmeyer operators.” Computers &amp; Mathematics with Applications 30.3-6 (1995): 1-14. <a href="https://doi.org/10.1016/0898-1221%2895%2900081-X"><strong>https://doi.org/10.1016/0898-1221%2895%2900081-X</strong></a> <a href="#fnref:55" class="reversefootnote" role="doc-backlink">&#8617;</a></p>
    </li>
  </ol>
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<p style='font-size:120%;font-weight:bold'><a href='https://peteroupc.github.io/approxtheory.pdf'>Download a PDF of this page</a></p><p>Of interest to the computer graphics and music community:</p><ul><li><a href='https://peteroupc.github.io/graphics.html'><b>Graphics for classic-style game development</b></a>.<li><a href='https://peteroupc.github.io/graphics.html#Building_a_Public_Domain_music_synthesis_library_and_instrument_banks'><b>MIDI music synthesis for classic-style games and apps</b></a>.<li><a href='https://peteroupc.github.io/classic-wallpaper'><b>Tileable wallpapers with limited colors and resolution</b></a>.<li><a href='https://peteroupc.github.io/graphicsapi.html'><b>Lean programming interfaces for classic graphics</b></a>.<li><a href='https://peteroupc.github.io/classic-wallpaper/docs/uielements.html'><b>Traditional user-interface graphics</b></a>.<li><a href='https://peteroupc.github.io/classic-wallpaper/docs/pixeltovector.html'><b>Converting images to vector graphics</b></a>.</ul><p>Open questions for math and probability experts:</p><ul><li><a href='https://peteroupc.github.io/requests.html'><b>Requests and open questions</b></a>.<li><a href='https://peteroupc.github.io/bernreq.html'><b>Open questions on the Bernoulli factory problem (the new-coins-from-old problem)</b></a>.<li><a href='https://peteroupc.github.io/requestsother.html'><b>Other open questions on probability</b></a>.<li><a href='https://peteroupc.github.io/sampling.html'><b>The sampling problem</b></a>.</ul></nav></body></html>