On convergence of rational and best rational approximations
type: Item
instance of: scholarly article
title: On convergence of rational and best rational approximations
zbMATH Open document ID: 0548.30030
DOI: 10.1016/0022-247X(84)90259-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: It is shown first that for a function f analytic everywhere except for the set of capacity zero and or arbitrary \(\eta >0\) (but not for \(\eta =0\), in general) interpolation at only \((1+\eta)\max\{M_ k,L_ k\}\) points (rather than the usual \(M_ k+L_ k+1\) points) is sufficient for a sequence of rational functions (of numerator degree \(\leq L_ k\), denominator degree \(\leq M_ k\), where \(1/\lambda\leq M_ k/L_ k\leq\lambda \) for some \(\lambda \geq 1\), and where \(M_ k,L_ k\to\infty )\) to converge in capacity as \(k\to\infty \), togethwer with all their derivatives. This impressive result, whose proof takes nearly six pages, generalizes previous results of Pommerenke and Wallin. It is applied to functions f that are additionally real on \({\mathbb{R}}\) and to their best real rational approximations on a series of real intervals. Further similar results concern the case where \(L_ k\to\infty \), \(M_ k/L_ k\to 0\). Some more recent related work is cited at the end of the paper.
reviewed by: Martin H. Gutknecht
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 3873672
zbMATH Keywords: best rational approximation
zbMATH Keywords: rational interpolation
author: Doron S. Lubinsky
MaRDI profile type: Publication
cites work: Q3235904
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cites work: Padé approximate as limits of rational functions of best approximation, real domain
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full work available at URL: 0022-247x(84)90259-2
OpenAlex ID: W2011080773
description: scientific article; zbMATH DE number 3873672