Conditional bounds and best \(L_{\infty}\)-approximations in probability spaces
type: Item
instance of: scholarly article
title: Conditional bounds and best \(L_{\infty}\)-approximations in probability spaces
zbMATH Open document ID: 0696.41025
DOI: 10.1016/0021-9045(89)90128-7
author: Carlos Matrán
author: Juan Antonio Cuesta
published in: Journal of Approximation Theory
publication date: 1989-01-01T00:00:00Z
review text: The paper studies \(L_{\infty}\)-approximation in a probability space (\(\Omega\),\(\sigma\),P) by means of \(\alpha\)-measurable random variables for \(\alpha\) \(\subset \sigma\), a \(\sigma\)-lattice. Besides characterizing the set of best \(L_{\infty}\)-approximations, the case of their attainment has also been taken up. The helpful notion of `conditional bounds' developed in the paper leads one to stronger results and simpler proofs. The results imply that if the best \(L_{\infty}\)-approximation is unique it could be computed via the Pólya algorithm. The paper provides an alternative simpler-and-wider technique of proving the existing results in the area as also corrects a mistake in Lemma 4 of \textit{R. B. Darst} [Proc. Am. Math. Soc. 81, 433-436 (1981; Zbl 0461.41023)].
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 60A10
zbMATH DE Number: 4140464
zbMATH Keywords: isotonic approximation
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Lebesgue-Radon-Nikodym derivatives
zbMATH Keywords: random variables
zbMATH Keywords: conditional bounds
zbMATH Keywords: Pólya algorithm
reviewed by: Ashok Sahai
MaRDI profile type: Publication
full work available at URL: 0021-9045(89)90128-7
OpenAlex ID: W2057836031
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description: scientific article; zbMATH DE number 4140464