Fourier coefficients of continuous functions and a class of multipliers
type: Item
instance of: scholarly article
title: Fourier coefficients of continuous functions and a class of multipliers
zbMATH Open document ID: 0607.42004
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74798
review text: If x is a bounded function on \({\mathbb{Z}}\), the multiplier with symbol \(x\) (denoted by \(M_ x)\) is defined by \((M_ xf){\hat{\;}}=x\hat f\), \(f\in L^ 2({\mathbb{T}})\). We give some conditions on \(x\) ensuring the ''interpolation inequality'' \(\| M_ xf\| _{L^ p}\leq C\| f\| ^{\alpha}_{L^ 1}\| M_ xf\| _{L^ q}^{1-\alpha}\) (here \(1<p<q\) and \(\alpha =\alpha (p,q,x)\) is between 0 and 1). In most cases considered \(M_ x\) fails to have stronger \(L^ 1\)-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets \(E\subset {\mathbb{Z}}\) every positive sequence in \(\ell ^ 2(E)\) can be majorized by the sequence \(\{| \hat f(n)|\}_{n\in E}\) for some continuous function \(f\) with spectrum in \(E\).
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A45
zbMATH DE Number: 3981761
zbMATH Keywords: multiplier
zbMATH Keywords: interpolation inequality
zbMATH Keywords: \(L^1\)-regularity properties
author: S. V. Kislyakov
MaRDI profile type: Publication
OpenAlex ID: W2333873663
cites work: Bilinear Forms on H ∞ and Bounded Bianalytic Functions
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Q4150143
cites work: Q3700300
cites work: Q3913635
cites work: Q3207619
cites work: Q4743237
cites work: Q4049615
cites work: Q3265794
cites work: Maximal inequalities of weak type
cites work: Einige Sätze und Fragestellungen über Fourier-Koeffizienten
cites work: Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets
cites work: Q5607484
cites work: Q3921478
cites work: Q3254471
DOI: 10.5802/AIF.1138
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 3981761