Rearrangement invariance and relations among measures of smoothness
type: Item
instance of: scholarly article
title: Rearrangement invariance and relations among measures of smoothness
zbMATH Open document ID: 1265.41061
author: Z. Ditzian
published in: Acta Mathematica Hungarica
publication date: 2012-09-18T00:00:00Z
review text: Let \(B\) be a Banach space of functions, defined on the \(d\)-dimensional torus or on the unit sphere in \(\mathbb{R} ^d\). Relations between the moduli of smoothness \(\omega ^r(f,t)_B\) and \(\omega ^{r+1}(f,t)_B\) of the sharp Marchaud and sharp lower estimate-type are shown to be satisfied for Banach spaces of functions that are not rearrangement invariant and for which translation or rotation are isometries. Moreover, analogies of the obtained results relating the rate of best approximation with \(\omega ^r(f,t)_B\), that is, the sharp converse inequality and the sharp Jackson inequality are established. For the case that \(B\) is a rearrangement invariant space, these results were obtained earlier by the author and others.
reviewed by: D. K. Ugulava
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 26D99
zbMATH DE Number: 6083364
zbMATH Keywords: moduli of smoothness
zbMATH Keywords: best approximation
zbMATH Keywords: sharp Marchaud inequality
zbMATH Keywords: sharp Jackson inequality
zbMATH Keywords: rearrangement invariance
MaRDI profile type: Publication
full work available at URL: s10474-011-0171-6
OpenAlex ID: W2062295432
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DOI: 10.1007/S10474-011-0171-6
description: scientific article; zbMATH DE number 6083364