Uniform approximation by polynomials on compacta of special form
type: Item
instance of: scholarly article
title: Uniform approximation by polynomials on compacta of special form
zbMATH Open document ID: 1279.41009
DOI: 10.3103/S0027132212030072
author: I. V. Beloshapka
published in: Moscow University Mathematics Bulletin
publication date: 2013-07-24T00:00:00Z
review text: Let \(K\) be a compact subset of the complex plane, and \(f\) a continuous, complex-valued function defined on \(K\). This paper presents lower bounds for the best uniform approximation (on \(K\)) to \(f\) by a polynomial of a specified finite degree. The estimates depend on variations of concepts such as: the existence of polynomials \(P\) and \(Q\), complex numbers \(v_j \in K\) such that \(|(f \, P)(Q(v_j))| = \|f\, P\|\) and constructing measures on \(\{v_j\}\) that annihilate the linear space of approximating polynomials. The results are compared to, and improve, known results in the literature, and they are applied to established systems such as Chebyshev systems. However, the presentation in this translation obfuscates the results and examples.
reviewed by: Daniel Wulbert
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 6190790
zbMATH Keywords: best approximation
zbMATH Keywords: complex-valued functions
zbMATH Keywords: lower bounds
MaRDI profile type: Publication
cites work: Chebyshev polynomials for Julia sets
cites work: On polynomials sharing preimages of compact sets, and related questions
full work available at URL: s0027132212030072
OpenAlex ID: W1998527327
description: scientific article; zbMATH DE number 6190790