On strong uniqueness in linear Chebyshev approximation
type: Item
instance of: scholarly article
title: On strong uniqueness in linear Chebyshev approximation
zbMATH Open document ID: 0537.41015
DOI: 10.1016/0021-9045(84)90109-6
author: H.-P. Blatt
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: Let Q be a compact subset of \({\mathbb{C}}\) and C(0) the set of all continuous functions f:\(Q\to {\mathbb{C}}\). A given function \(f\in C(Q)\) is uniformly approximated by elements of an n-dimensional Haar subspace V of C(Q). \textit{D. J. Newman} and \textit{H. Shapiro} [Duke Math. J. 30, 673-681 (1963; Zbl 0116.045)] have shown that the best approximation \(\tilde v\in V\) to f is in general not strongly unique. The main result of the present paper can be stated as follows: There exists an open and dense subset of C(Q) such that all functions of this subset have a strongly unique best approximation iff Q has at most n isolated points. Hence, in this case, strong uniqueness is a generic property.
Mathematics Subject Classification ID: 41A52
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 3853673
zbMATH Keywords: Chebyshev approximation
zbMATH Keywords: strong uniqueness
zbMATH Keywords: strongly unique best approximation
zbMATH Keywords: isolated points
MaRDI profile type: Publication
full work available at URL: 0021-9045(84)90109-6
OpenAlex ID: W1979572580
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description: scientific article; zbMATH DE number 3853673