On projections in \(L_ 1\) and \(L_{\infty}\)
type: Item
instance of: scholarly article
title: On projections in \(L_ 1\) and \(L_{\infty}\)
zbMATH Open document ID: 0601.41036
DOI: 10.1007/BF01890037
author: Boris Shekhtman
published in: Constructive Approximation
publication date: 1985-01-01T00:00:00Z
review text: Let \(X_ n\) be a nested family of n-dimensional subspaces of \(L_{\infty}\). Let \(P_ n\) be a sequence of projections from either \(L_ 1\) or \(L_{\infty}\) onto \(X_ n\). Suppose the projections are written in their integral form \((P_ nf)(t)=\int K_ n(s,t)f(s)ds.\) It is proved that if \(\| K_ n-K_{n-1}\|_{\infty}<o(\log (n)),\) then \(\sup \| P_ n\| =\infty\). The essence of the paper is a complicated construction proving a lower bound for the average of the integrals (double) of the kernels \(K_ n\). This inequality is then applied to prove three theorems including the one above. This application extends known results.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 41A45
zbMATH DE Number: 3969290
zbMATH Keywords: nested family of n-dimensional subspaces
zbMATH Keywords: projections
MaRDI profile type: Publication
cites work: An estimation of the Lebesgue functions of biorthogonal systems with an application to the non-existence of some bases in C and $L^{1}$
cites work: Q4084169
full work available at URL: https://doi.org/10.1007/bf01890037
OpenAlex ID: W3005870245
description: scientific article; zbMATH DE number 3969290