A counterexample on global Chebyshev approximation
type: Item
instance of: scholarly article
title: A counterexample on global Chebyshev approximation
zbMATH Open document ID: 0651.41014
DOI: 10.1016/0021-9045(87)90023-2
author: Guohui Feng
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: Let Y be a non-empty subset of a B-space X and \(x\in X\). Then \(y\in Y\) is called a best approximation to x if \(\| x-y\| =dist(x,Y)\). Y is said to be proximinal if each \(x\in X\) has at least one best approximation in Y. If A,Y are subsets of a B-space X then the Chebyshev radius and the Chebyshev centre of A relative to Y are defined respectively as follows: \(r_ Y(A)=\inf_{y\in Y}\sup_{a\in A}\| a-y\|\) and \(E_ Y(A)=\{y\in Y:\sup_{a\in A}\| a-y\| =r_ Y(A)\}\). If \(Y=X\) we simply speak about the Chebyshev radius r(A) and the Chebyshev centre E(A) of A, respectively. It has been shown by Franchetti and Cheney that the Chebyshev centre of any bounded subset of C(S) (where S is a compact Hausdorff space) is non- empty. This is equivalent to answering whether or not Y is proximinal in C(S\(\times T)\) for each compact Hausdorff space T. In this paper the author constructs a counterexample to answer this question in the negative.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4061853
zbMATH Keywords: proximinal
zbMATH Keywords: Chebyshev radius
zbMATH Keywords: Chebyshev centre
reviewed by: Constantine G. Lascarides
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90023-2
OpenAlex ID: W1993442600
Wikidata QID: Q124832544
cites work: THE CHARACTERIZATION OF BEST APPROXIMATIONS IN TENSOR-PRODUCT SPACES
cites work: Restricted Centers in C(Ω)
description: scientific article; zbMATH DE number 4061853