Polynomial approximation on polytopes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.41009
author: Vilmos Totik
publication date: 2014-10-17T00:00:00Z
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 6357635
zbMATH Keywords: polynomials
zbMATH Keywords: several variables
zbMATH Keywords: approximation
zbMATH Keywords: moduli of smoothness
zbMATH Keywords: \(K\)-functionals
zbMATH Keywords: strong inequalities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1090/memo/1091
OpenAlex ID: W4246453428
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DOI: 10.1090/MEMO/1091
published in: Memoirs of the American Mathematical Society
title: Polynomial approximation on polytopes
review text: A subset \(K\) of \(\mathbb{R}^{d}\) is said to be simple polytope if there are precisely \(d\) edges at every vertex of \(K\). If \(E_{n}(f)_{K}\) denotes the error of the best approximation of a continuous function \(f\) in \(K\) by polynomials of degree at most \(n\), it was proven in [\textit{Z. Ditzian} and \textit{V. Totik}, Moduli of smoothness. New York etc.: Springer-Verlag (1987; Zbl 0666.41001)], that you have the estimate \(E_{n}(f)_{K}\leq M (\omega_{K}^{r} (f, \frac{1}{n}) + n^{-r}\|f\|_{K})\). Here \(\omega_{K}^{r} (f, \frac{1}{n})\) denotes the \(r\)-modulus of smoothness of \(f\) that will be defined below.NEWLINENEWLINENEWLINEIn the last 25 years it remains open the problem of the validity of such a kind of estimate for non simple polytopes. Notice that the second term in the right hand side of the previous inequality is usually dominated by the first one. The aim of this monograph is to cover this gap by proving for \(d\)-dimensional convex polytopes, i.e., closed sets in \(\mathbb{R}^{d}\), with an inner point, such that they are the convex hull of finitely many points, the following Jackson-type or direct theorem:NEWLINENEWLINETheorem. Let \(K\) a \(d\)-dimensional convex polytope and \(r=1, 2, \dots\). Then, for \(n\geq rd\), \(E_{n}(f)_{K}\leq M \omega^{r}_{K} (f, \frac{1}{n})\), where \(M\) depends only on \(K\) and \(r\).NEWLINENEWLINENEWLINEThe modulus of smoothness on \(K\) is defined as follows. Let \(f\) be a continuous function on \(K\). Its \(r\)-th symmetric difference in a direction \(e,\) i.e., a vector in \(\mathbb{R}^{d}\) with \(\|e\|=1\), is \(\Delta^{r}_{he}f(x)= \sum _{k=0}^{r}(-1)^{k} \frac{r(r-1) \cdots (r-k+1)}{k!} f(x+ (\frac{r}{2}-k) he)\) assuming that this is \(0\) if \(x + \frac{r}{2} he\) or \(x- \frac{r}{2} he\) does not belong to \(K\). Then \(\omega_{K}^{r} (f, \delta)= \sup_{\|e\|=1} \sup_{h\leq \delta} \|\Delta^{r}_{h\tilde{d}_{K}(e, x) e} f(x)\|_{K}.\) Here \(\tilde{d}_{K}(e,x)\) is the so-called normalized dist…
reviewed by: Francisco Marcellán
description: scientific article; zbMATH DE number 6357635