The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials
type: Item
instance of: scholarly article
title: The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials
zbMATH Open document ID: 0752.42001
DOI: 10.1016/0022-247X(91)90344-Y
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(T\) be a given trigonometric polynomial of degree \(n\) and \(p\) be a number such that \(1<p<+\infty\). The well-known Marcinkiewicz-Zygmund inequality [see \textit{A. Zygmund}: ``Trigonometric series'', Vol. II (1959; Zbl 0085.056), p. 28] provides an estimate for \((*) [\int_ 0^{2\pi}| T(x)|^ pdx]^{1/p}\) in terms of a sum involving the values of \(T\) at certain equispaced points in \([0,2\pi]\). The author generalises this inequality to trigonometric polynomials of a certain class in such a way that \((*)\) is estimated in terms of the values of the polynomial and its derivatives at prescribed equispaced points. Such a generalisation is then used to prove the mean convergence of the Hermite and some other interpolants.
reviewed by: H. P. Dikshit
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42A15
zbMATH DE Number: 57724
zbMATH Keywords: trigonometric polynomial
zbMATH Keywords: Marcinkiewicz-Zygmund inequality
zbMATH Keywords: mean convergence
author: Yuan Xu
MaRDI profile type: Publication
cites work: Q3961281
cites work: Sur l'approximation d'une fonction périodique et de ses dérivées successives par un polynôme trigonométrique et par ses dérivées successives
cites work: Замечания об ихтерполировании
cites work: Quadrature Sums Involving pth Powers of Polynomials
cites work: Convergence of lacunary trigonometric interpolation on equidistant nodes
cites work: Q3950976
full work available at URL: 0022-247x(91)90344-y
OpenAlex ID: W2077588304
description: scientific article; zbMATH DE number 57724