Weighted polynomial approximation in the complex plane
type: Item
instance of: scholarly article
title: Weighted polynomial approximation in the complex plane
zbMATH Open document ID: 0920.30028
published in: Constructive Approximation
publication date: 1999-09-09T00:00:00Z
review text: The paper is concerned with pairs \((G,W)\), where \(G\) is a bounded open set in the complex plane which can be represented as a finite or countable union of disjoint simply connected regions, and \(W(z)\), the weight function, is analytic in \(G\) with \(W(z)\neq 0\). The pair \((G,W)\) has the approximation property if any analytic function \(f\) in \(G\) can be represented on any compact \(E\subset G\) as the uniform limit of weighted polynomials \(W^nP_n\) as \(n\rightarrow \infty\), with polynomials \(P_n\) of degree \(\leq n\). The authors' main result is the criterion: The pair \((G,W)\) has the approximation property if and only if the harmonic function \(\log | W(z)| \) can be represented (up to an additive constant) as the logarithmic potential of a unit measure \(\mu\) with support in \(\partial G\). This result is applied to various classical weights and gives explicit criteria for these approximation problems. The measure \(\mu\) is related to the solution of a minimal weighted energy problem.
reviewed by: Rudolf Wegmann
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1268075
zbMATH Keywords: weighted polynomials
zbMATH Keywords: weighted energy problem
zbMATH Keywords: logarithmic potential
author: Igor E. Pritsker
author: Richard S. Varga
MaRDI profile type: Publication
full work available at URL: s003659900086
OpenAlex ID: W2081574909
DOI: 10.1007/S003659900086
description: scientific article; zbMATH DE number 1268075