An embedding theorem for functions whose Fourier transforms are weighted square summable
type: Item
instance of: scholarly article
title: An embedding theorem for functions whose Fourier transforms are weighted square summable
zbMATH Open document ID: 0807.46031
DOI: 10.4171/ZAA/506
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1994-09-01T00:00:00Z
review text: Summary: Our main result is, by elementary means only, an embedding theorem for functions \(f\) whose Fourier transforms \(\widetilde{f}\) are weight square summable, i.e. \(\int_{\mathbb{R}^ n} | \widetilde{f} (\lambda)|^ 2 w(\lambda) d\lambda<\infty\), where the weight function \(w: \mathbb{R}^ n\to (0,\infty)\) satisfies the Kolmogorov condition \(\int_{\mathbb{R}^ n} w(\lambda)^{-1} d\lambda <\infty\). The necessary and sufficient condition on a non-negative Borelian measure on \(\mathbb{R}^ n\) is given, for the inequality \(\int_{\mathbb{R}^ n} | f(t)|^ 2 \mu(dt)\leq A\int_{\mathbb{R}^ n} |\widetilde{f} (\lambda)|^ 2 w(\lambda)d\lambda\) to hold for every such function \(f\), with a constant \(A<\infty\) not depending on \(f\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 627642
zbMATH Keywords: weighted functional spaces
zbMATH Keywords: embedding theorem
zbMATH Keywords: Fourier transforms
author: V. Katsnelson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/506
OpenAlex ID: W2565731025
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description: scientific article; zbMATH DE number 627642