\((T,E)\)-Korovkin closures in normed spaces and BKW-operators
type: Item
instance of: scholarly article
title: \((T,E)\)-Korovkin closures in normed spaces and BKW-operators
zbMATH Open document ID: 0865.47008
author: S.-E. Takahasi
published in: Journal of Approximation Theory
publication date: 1997-07-13T00:00:00Z
review text: Let \(X\) and \(Y\) be normed spaces, \(S\) a subset of \(X\), \(E\) a bounded subset of the dual \(Y^*\), and \(T\in B(X,Y)\) (i.e., \(T\) is a bounded linear operator from \(X\) to \(Y\)). The author defines \(T\) to be a BKW (for Bohman, Korovkin, and Wulbert) operator (relative to \(X\), \(Y\), \(S\), and \(E\)) if for every net \(T_\lambda\in B(X,Y)\) such that \(\lim_\lambda|T_\lambda|=|T|\) and \(\sup_s \lim_\lambda\sup_{\phi\in E}|\phi(T_\lambda(s)- T(s)|=0\), it follows that \(\lim_\lambda\sup_{\phi\in E}|\phi(T_\lambda(x)- T(x)|=0\) for every \(x\in X\). The functions in \(S\) are termed test functions. Clearly, a small set of test functions means a severe restriction on the operator \(T\). The author illustrates this in Theorem 2.1, where the set \(S\) is \(\{1,x\}\) and \(\{1,x,x^2\}\), respectively (here \(x\) is the identity function on \([0,1])\). In both cases there are extremely few BKW operators, which the author describes completely.
reviewed by: H. Bercovici
Mathematics Subject Classification ID: 47A58
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 803699
zbMATH Keywords: Bohman
zbMATH Keywords: Korovkin
zbMATH Keywords: Wulbert
zbMATH Keywords: BKW operators
MaRDI profile type: Publication
full work available at URL: jath.1995.1083
OpenAlex ID: W2093130195
DOI: 10.1006/JATH.1995.1083
description: scientific article; zbMATH DE number 803699