On a theorem of Marcinkiewicz and Zygmund
type: Item
instance of: scholarly article
title: On a theorem of Marcinkiewicz and Zygmund
zbMATH Open document ID: 0922.42018
DOI: 10.1007/BF02317779
author: S. A. Kirillov
published in: Mathematical Notes
publication date: 1999-04-22T00:00:00Z
review text: The author proves the following interesting theorem. There exists an orthogonal system of functions \(\{\varphi_n(x)\}\), \(\varphi_n\in L_\infty[0,1]\) such that for any \(q>2\) there is a sequence \(\{c_n\}\) for which \[ \sum^\infty_{n=1}| c_n|^qn^{q-2}\|\varphi_n\|_\infty^{q-2}<\infty, \] the series \(\sum c_n\varphi_n(x)\) is convergent in \(L_2[0,1]\), but its sum does not belong to \(L_q[0,1]\). This result shows that the classical Marcinkiewicz-Zygmund theorem is sharp.
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 1262561
zbMATH Keywords: orthogonal system
zbMATH Keywords: Marcinkiewicz-Zygmund theorem
reviewed by: László Leindler
MaRDI profile type: Publication
cites work: Interpolation of Operators with Change of Measures
cites work: On a Theorem of Paley
cites work: Properties of the Coefficients of Orthonormal Sequences
cites work: Some generalizations of the Hardy-Littlewood-Paley theorem
cites work: Q5601561
full work available at URL: https://doi.org/10.1007/bf02317779
OpenAlex ID: W2027223355
description: scientific article; zbMATH DE number 1262561