Stability of best rational Chebyshev approximation
type: Item
instance of: scholarly article
title: Stability of best rational Chebyshev approximation
zbMATH Open document ID: 0727.41022
DOI: 10.1016/0021-9045(90)90009-F
author: Bruno Brosowski
author: Claudia S. Guerreiro
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
review text: The authors study best Chebyshev approximation from the standpoint of optimization. Consider \((y,z)\in {\mathbb{R}}^ n\times {\mathbb{R}}\) and pose the problem minimize \(p(v,z)=z\), subject to (i) \(v\in U\), (ii) for all (\(\eta,s)\in T:\) \(\eta \frac{<B_{s,v}>}{<C_{s,v}>}-\gamma (\eta,s)z\leq \eta x(s),\) where \(S\neq \emptyset\) is compact Hausdorff space, \(B,C: S\to R^ n\) are continuous functions such that the open convex set \(U=\cap_{s\in S}\{v\in R^ n| <C(s),v>>0\}\) is nonempty, \(\gamma: T=\{-1,1\}\times S\to R\) is a nonnegative continuous function such that \(\gamma(-1,s)+ \gamma(1,s)>0\) for all \(s\in S\) and \(x: S\to R\) is continuous. The paper then goes on introducing feasible sets, the minimal value for \(p(v,z)\), the solutions attaining this value, the solutions that can be written as a generalized rational function and attain this minimal value and the class of problems \((B,C,\gamma,x)\) for which the solvability set is nonempty and a condition for a problem to belong to this class (the Slater condition). Then a number of results on lower/upper (semi)continuity of the set theoretic mappings connected with the feasible sets, minimal values and minimal solution sets are given. The style of the paper is rather compact but - after mastering the definitions and concepts that play a role - it is very interesting and quite readable.
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 4200864
zbMATH Keywords: best Chebyshev approximation
zbMATH Keywords: optimization
zbMATH Keywords: feasible sets
zbMATH Keywords: minimal values
zbMATH Keywords: minimal solution sets
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
cites work: Q3968042
cites work: Q3968041
cites work: Radial continuity of set-valued metric projections
cites work: Parametric approximation
cites work: Q3217182
cites work: Conditions for the uniqueness of best generalized rational Chebyshev approximation to differentiable and analytic functions
cites work: Q3733418
cites work: An extension of strong uniqueness to rational approximation
cites work: Q3027568
cites work: Uniform approximation with constraints
cites work: On the continuity of rational approximation operators
cites work: Continuity of the best approximation operator for restricted range approximations
cites work: Tschebyscheff-Approximationen in kleinen Intervallen. I; II: Stetigkeitssätze für gebrochen rationale Approximationen
cites work: On Approximations by Polynomials Having Restricted Ranges. II
cites work: On Approximation by Polynomials Having Restricted Ranges
cites work: On the rational Tschebyscheff operator
full work available at URL: 0021-9045(90)90009-f
OpenAlex ID: W2038426826
description: scientific article; zbMATH DE number 4200864