Polynomial approximation in convex domains
type: Item
instance of: scholarly article
title: Polynomial approximation in convex domains
zbMATH Open document ID: 1401.41004
published in: Journal of Approximation Theory
publication date: 2018-09-27T00:00:00Z
review text: Let \(G\) be a compact convex domain in \(\mathbb{R}^{d}\) and \(\Gamma \subset \partial G\) denotes the set of its conic points (\(x \in \Gamma\) if there are \(d\) linearly independent supporting hyperplanes of \(G\) through \(x\)). For \(f \in C(G)\) and \(t>0,\) the \(k\)th order modulus of smoothness is given by \[ \omega_{k}(f;t) = \sup_{| h | \leq t} \, \sup_{x \in G_{kh}} \, | \Delta_{h}^{k} f(x) | , \] where \(G_{kh} = \{ x : [x,x+kh] \subset G \},\) \(h \in \mathbb{R}^{d}\) and \[ \Delta_{h}^{k} f(x) = \sum_{j=0}^{k} \, (-1)^{k-j} \binom{k}{j} f(x+jh). \] The main result of the paper is the following Theorem. Let \(f \in C^{l}(G)\) and \(\Gamma \subset \partial G,\) \(l \geq 0,\) \(k,n \in \mathbb{Z}_{+}\) satisfy the conditions \(k \geq 1,\) \(n \geq k+l-1\) and \(\operatorname{card} \Gamma < \infty.\) Then there is a linear operator \(R_{n} : C(G) \to \mathcal{P}_{n}\) such that \[ | f(x) - R_{n}f(x) | \leq c \, \Delta_{n}(x)^{l} \sup_{| \alpha | = l} \, \omega_{k}\left( D^{\alpha}f ; \Delta_{n}(x) \right), \quad x \in G; \] \(\mathcal{P}_{n}\) is the space of polynomials in \(x \in \mathbb{R}^{d}\) of degree \(n;\) \(c>0\) is a constant independent of \(x, f, n;\) \(\Delta_{n}(x) = \frac{\sqrt{\mathrm{dist} (x,\Gamma)}}{n+1} + \frac{1}{(n+1)^{2}}\) and \(| \alpha | = \sum_{i=1}^{d} \, \alpha_{i}\) for \(\alpha = (\alpha_{1}, \dots, \alpha_{d}) \in \mathbb{Z}^{d}_{+}.\)
reviewed by: Zoltán Finta
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 6944961
zbMATH Keywords: polynomial approximation
zbMATH Keywords: convex domain
zbMATH Keywords: conic point
zbMATH Keywords: modulus of continuity
zbMATH Keywords: polynomial operator
author: Yuri A. Brudnyi
MaRDI profile type: Publication
full work available at URL: j.jat.2018.08.001
OpenAlex ID: W2885342254
cites work: Q5512298
cites work: Q5551610
cites work: Q5602381
cites work: Q3905935
cites work: A practical guide to splines
cites work: Q4112374
cites work: Q3237285
cites work: Q3256417
cites work: Über die Approximation reeller stetigen Funktionen durch gewöhnliche Polynome
cites work: Q5551609
cites work: Q4118037
cites work: Q4064526
cites work: Q5607484
cites work: Two theorems on the approximation of functions by algebraic polynomials
cites work: Q3246572
cites work: On functions with bounded \(n\)-th differences
DOI: 10.1016/J.JAT.2018.08.001
description: scientific article; zbMATH DE number 6944961