Linear problems (with extended range) have linear optimal algorithms
type: Item
instance of: scholarly article
title: Linear problems (with extended range) have linear optimal algorithms
zbMATH Open document ID: 0637.41024
DOI: 10.1007/BF02188168
author: Edward W. Packel
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137138
review text: Let \(F_ 1\) and \(F_ 2\) be normed linear spaces and \(S: F_ 0\to F_ 2\) a linear operator on a balanced subset \(F_ 0\) of \(F_ 1\). If N denotes a finite dimensional linear information operator on \(F_ 0\), it is known that there need not be a linear algorithm \(\phi\) : N(F\({}_ 4)\to F_ 2\) which is optimal in the sense that \(\| \phi (N(f))- S(f)\|\) is minimized. We show that the linear problem defined by S and N can be regarded as having a linear optimal algorithm if we allow the range of \(\phi\) to be extended in a natural way. The result depends upon imbedding \(F_ 2\) isometrically in the space of continuous functions on a compact Hausdorff space X. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.
Mathematics Subject Classification ID: 41A45
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 68W99
zbMATH DE Number: 4038424
zbMATH Keywords: linear algorithm
zbMATH Keywords: optimal algorithm
MaRDI profile type: Publication
cites work: Best approximation of analytic functions from information about their values at a finite number of points
cites work: Q3874734
cites work: Q3883494
description: scientific article; zbMATH DE number 4038424