Best monotone approximation in \(L_{\infty}[0,1]\)
type: Item
instance of: scholarly article
title: Best monotone approximation in \(L_{\infty}[0,1]\)
zbMATH Open document ID: 0545.41022
DOI: 10.1016/0021-9045(84)90053-4
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: If f(x) is a Lebesgue measurable function on [0,1] and \(p>1\), let \(f_ p(x)\) be the unique best \(L_ p\)-approximant to f(x) by non-decreasing functions on [0,1]. If \(\lim_{p\to \infty}f_ p(x)\equiv f_{\infty}(x)\) exists a.e., then \(f_{\infty}(x)\) is a best \(L_{\infty}\)-approximant to f(x) by non-decreasing functions. In this case we say that the Polya algorithm converges and \(f_{\infty}(x)\) is a best best \(L_{\infty}\)-approximant. Darst and Sahab have shown that if f(x) is quasi-continuous, then the Polya algorithm converges. Darst, Legg and Townsend have shown that if f(x) is only assumed Lebesgue measurable, then the algorithm may fail to converge. In this paper, we show that the condition that f(x) be quasi-continuous can be relaxed to the condition that f(x) can be uniformly approximated by simple measurable functions where one-sided limits need only exist at a few select points. The authors believe the construction of \(f_{\infty}(x)\) as given in this paper gives a clearer picture of what \(f_{\infty}(x)\) is, even when f(x) is continuous.
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 3867915
zbMATH Keywords: Polya algorithm
zbMATH Keywords: best best \(L_{\infty}\)-approximant
author: David A. Legg
author: Douglas W. Townsend
MaRDI profile type: Publication
full work available at URL: 0021-9045(84)90053-4
OpenAlex ID: W2046049166
cites work: The Polya algorithm in \(L_ \infty\) approximation
cites work: On projections and monotony in \(L_p\)-spaces
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description: scientific article; zbMATH DE number 3867915