The Paley-Wiener theorem with general weights
type: Item
instance of: scholarly article
title: The Paley-Wiener theorem with general weights
zbMATH Open document ID: 0722.32003
DOI: 10.1016/0022-247X(90)90225-5
author: T. G. Genchev
author: H. P. Heinig
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: The authors consider weight functions u(x), v(x) in \({\mathbb{R}}^ n\), \(p\geq 1\), \(q\geq 1\) that satisfy the condition \(F^ p_ q: \| \hat f\|_{q,u}\leq C\| f\|_{p,v}\) for any simple function on \({\mathbb{R}}^ n\). Here \(\hat f\) is the Fourier transform of f, \(\| f\|_{p,w}=\| w^{1/p}f\|_ p\) and \(\| \cdot \|_ p\) is the usual \(L_ p\)-norm on \({\mathbb{R}}^ n.\) The main result of the paper is the following statement (Theorem 3): Let the weight functions u,v satisfy the condition \(F^ p_ q\), u(x) be even, v(x) be such that \(v(x)\geq c>0\) for \(| x| \geq N>0\). Let also an entire function f(z), \(z\in {\mathbb{C}}^ n\), be such that \(\| f\|_{p,v}<\infty\) and \(| f(z)| <A_{\epsilon}e^{(\sigma +\epsilon)| z|}\), \(\epsilon >0\). Then a measurable function g(x), \(x\in {\mathbb{R}}^ n\), exists such that \(\| g\|_{q,u}<\infty\), supp \(g\subset \{| t| \leq \sigma \}\) and \[ f(z)=\int_{{\mathbb{R}}^ n}e^{i<z,t>}g(t)dt. \] A number of corollaries are obtained. They generalize results of previous articles of the authors devoted to the same problems. There are also conditions on some special class of pairs (u,v) which are sufficient for \(F^ p_ q\).
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 4190129
zbMATH Keywords: function of exponential type
zbMATH Keywords: Fourier transform
zbMATH Keywords: weight functions
MaRDI profile type: Publication
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cites work: Notes on the Theory and Application of Fourier Transforms, III, IV, V, VI, VII
full work available at URL: 0022-247x(90)90225-5
OpenAlex ID: W2093649085
description: scientific article; zbMATH DE number 4190129