On best partial bases
type: Item
instance of: scholarly article
title: On best partial bases
zbMATH Open document ID: 0606.41036
DOI: 10.1016/0021-9045(86)90022-5
author: James T. Lewis
author: Oved Shisha
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: The partial basis problem is to determine a subsequence of given length n of a given basis \(h_ 1,...,h_ N\) in a normed linear space X for which best approximation to a given f in X is closest. Conditions guaranteeing that the initial segment \(h_ 1,...,h_ n\) (or terminal segment \(h_{N-n+1},...,h_ N)\) is the unique best partial basis are obtained. The results are closely connected with the theory of Chebyshev systems and extended complete Chebyshev systems. One of the nicest results is the following: Theorem. Let \(0<a<b<\infty\); n, N integers, \(1\leq n<N\). Let f be a real function, positive and continuous on [a,b] and assume that for \(k=1,2,...,n\), \((-1)^ kf^{(k)}(x)>0\) throughout (a,b). Let \(1\leq p\leq \infty\). Then \(1,x,...,x^{n-1}\) is the unique best partial basis of \(1,x,...,x^{N-1}\) of length n to approximate f in the \(L^ p(a,b)\) norm.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 3979697
zbMATH Keywords: partial basis problem
zbMATH Keywords: Chebyshev systems
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90022-5
OpenAlex ID: W1964313857
cites work: Q5543516
cites work: Disconjugacy
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cites work: Tchebycheff systems and best partial bases
cites work: What is the Laplace Transform?
description: scientific article; zbMATH DE number 3979697