Weighted rational approximation in the complex plane
type: Item
instance of: scholarly article
title: Weighted rational approximation in the complex plane
zbMATH Open document ID: 0932.30036
DOI: 10.1016/S0021-7824(01)80008-8
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-03-08T00:00:00Z
review text: Given a domain \(G\subset\mathbb{C}\), \(f\) analytic in \(G\), a weight function \(W\) also analytic in \(G\) and \(W(z)\neq 0\) in \(G\), and \(\gamma\) with \(0\leq\gamma\leq 1\), the authors study the rational approximation property of \(W\): There exist rational functions \(R_i=P_{m_i}/Q_{n_i}\) where \(P,Q\) are polynomials of degree \(\leq m_i\) and \(\leq n_i\), respectively, such that \(m_i/(m_i+n_i) \to \gamma\) \((1\to\infty)\) and \(\|f-W^{m_i+ n_i}\cdot R_i\|_E\to 0\) \((i\to \infty)\). The main result gives necessary and sufficient conditions for \(W\), expressed in terms of potential theoretic quantities, so that \(W\) has the rational approximation property. The cases \(W(z)=z^\alpha\) \((\alpha>0)\) with \(G=\{z:|z-a|<r\}\) \((a>0\) and \(r<a)\), and \(W(z)=e^{-z}\) with \(G=\{z: |z|<r\}\) \((r>0)\) are treated as special cases, as well as \(W(z)=e^{-z}\) with \(G\) a generalized Szegö domain. Earlier results by the authors [Constructive Approximation 14, 475-492 (1998; Zbl 0920.30028)] dealt with weighted polynomial approximation in \(\mathbb{C}\), while \textit{P. Borwein}, \textit{E. A. Rakhmanov} and \textit{E. B. Saff} [ibid. 12, No. 2, 223-240 (1996; Zbl 0870.41010)] considered weighted rational approximation on the real line.
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 1344446
zbMATH Keywords: complex approximation
zbMATH Keywords: rational approximation
author: Igor E. Pritsker
author: Richard S. Varga
reviewed by: Dieter Gaier
MaRDI profile type: Publication
cites work: Incomplete rational approximation in the complex plane
cites work: Rational approximation with varying weights I
cites work: Q3919971
cites work: Q3919041
cites work: The role of the endpoint in weighted polynomial approximation with varying weights
cites work: Q5666837
cites work: Q4163892
cites work: Polynomial approximation with varying weights on compact sets of the complex plane
cites work: The Szego curve, zero distribution and weighted approximation
cites work: Weighted polynomial approximation in the complex plane
cites work: Rational approximation with varying weights. II
cites work: Q4360405
cites work: Q4195371
cites work: On the zeros and poles of Padè approximants to \(e^z\). III
cites work: Weighted polynomial approximation for weights with slowly varying extremal density
cites work: Q3259806
cites work: Asymptotics for the zeros and poles of normalized Padé approximants to \(e^ z\)
cites work: Q5762274
cites work: Q5620934
description: scientific article; zbMATH DE number 1344446