A Korovkin theorem for abstract Lebesgue spaces
type: Item
instance of: scholarly article
title: A Korovkin theorem for abstract Lebesgue spaces
zbMATH Open document ID: 0969.41015
published in: Journal of Approximation Theory
publication date: 2001-09-18T00:00:00Z
review text: Let \(E\) be an AL-space (a Kakutani abstract L-space) with a generalized weak unit \(\{e_n\}\). The theorem proved here is a little stronger then the following. Theorem: Let \(\{T_i\}\) be a sequence of norm one operators defined on \(E\). Let \(L=\{ f\in E: T_if\) converges weakly to \(f\}\), and \(N=\{f\in E: T_if\) converges in norm to \(f\}\). If, for each \(i\), \(T_ie_n\) converges in norm to \(e_i\), then (i) \(N=L\), and (ii) \(N\) is a lattice. Even when restricted to basic spaces such as \(E=L_1[0,1]\), this theorem generalizes stated results in the literature. \{Reviewer's comment: The author compares a result here with the reviewer's work [J. Approximation Theory 1, 381-390 (1968; Zbl 0167.12904)]. To the reviewer's chagrin, it has repeatedly been pointed out that the theorem there stating that \(\{1, x, x^2\}\) is a Korovkin set is true when the function \(x^2\) is deleted from the set. No changes to the proof are required\}.
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1417549
zbMATH Keywords: weak convergence
zbMATH Keywords: lattice
zbMATH Keywords: contractive projection
author: Peter F. Renaud
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
full work available at URL: jath.1999.3377
OpenAlex ID: W1993353405
cites work: Q5331549
cites work: A Korovkin Type Theorem on Weak Convergence
cites work: Q4042131
cites work: Convergence of operators and Korovkin's theorem
DOI: 10.1006/JATH.1999.3377
description: scientific article; zbMATH DE number 1417549