Approximation related to quotient functionals
type: Item
instance of: scholarly article
title: Approximation related to quotient functionals
zbMATH Open document ID: 1194.41036
author: T. Teuber
author: S. Setzer
author: Gabriele Steidl
author: Guido Moerkotte
published in: Journal of Approximation Theory
publication date: 2010-05-11T00:00:00Z
review text: The best approximation of componentwise positive vectors or positive continuous functions \(f\) is examined by linear combinations \(\hat f = \sum _j \alpha _j \varphi _j\) of given vectors or functions \(\varphi _j\) with respect to functionals \(Q_p, 1\leq p\leq \infty \), involving quotients \(\max\{f/\hat f ,\hat f /f\}\) rather than differences \(|f-\hat f|\). The existence of a best approximating function under mild conditions on \(\{\varphi _j\}_{j=1}^n\) is verified. For discrete data, a best approximating function with respect to \(Q_p\), \(p=1,2,\infty \), is computed by second order cone programming. Special attention is paid to the \(Q_\infty \) functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional \(Q_\infty \) an equivalent characterization of the best approximation by using its extremal set is given. Then this characterization is applied to prove the uniqueness of the best \(Q_\infty \) approximation for Chebyshev sets \(\{\varphi _j\}_{j=1}^n\).
reviewed by: Juri M. Rappoport
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 5706948
zbMATH Keywords: best approximation
zbMATH Keywords: polynomial approximation
zbMATH Keywords: Chebyshev sets
zbMATH Keywords: convex optimization
zbMATH Keywords: second order cone programming
describes a project that uses: Mosek
MaRDI profile type: Publication
full work available at URL: j.jat.2009.08.009
OpenAlex ID: W2073481046
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DOI: 10.1016/J.JAT.2009.08.009
description: scientific article; zbMATH DE number 5706948