Functions and their Fourier transforms with supports of finite measure for certain locally compact groups
type: Item
instance of: scholarly article
title: Functions and their Fourier transforms with supports of finite measure for certain locally compact groups
zbMATH Open document ID: 0658.43003
DOI: 10.1016/0022-1236(88)90035-3
author: John Price
author: Alladi Sitaram
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let G be a locally compact group, and f an integrable function on G. The Fourier transform of f, an operator-valued function on the unitary dual \(\hat G\) of G is written \(\hat f.\) We let \(A_ f=\{x\in G:\) \(f(x)\neq 0\}\) and \(B_ f=\{\lambda \in \hat G:\) \(\hat f(\lambda)\neq 0\}\). The authors investigate the significance of the two inequalities \(m(A_ f)<m(G)\) and \(\mu (B_ f)<\mu (\hat G)\), where m is Haar measure on G and \(\mu\) is Plancherel measure on \(\hat G,\) or related inequalities. It is shown that, for many groups G, if these two conditions hold, then \(f=0\) a.e., generalizing a theorem of \textit{M. Benedicks} [J. Math. Anal. Appl. 106, 180-183 (1985; Zbl 0576.42016)] for \({\mathbb{R}}^ n\). The groups G considered include compact Lie groups, the group of Euclidean motions of \({\mathbb{R}}^ 2\), the \(``ax+b\)-group'', the Heisenberg group, and SL(2,\({\mathbb{R}})\).
Mathematics Subject Classification ID: 43A30
zbMATH DE Number: 4075768
zbMATH Keywords: uncertainty principle
zbMATH Keywords: locally compact group
zbMATH Keywords: integrable function
zbMATH Keywords: Fourier transform
zbMATH Keywords: unitary dual
zbMATH Keywords: Haar measure
zbMATH Keywords: Plancherel measure
zbMATH Keywords: compact Lie groups
zbMATH Keywords: group of Euclidean motions
zbMATH Keywords: Heisenberg group
reviewed by: Michael G. Cowling
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90035-3
OpenAlex ID: W2066238054
cites work: On support properties of Lsup(p)-functions and their Fourier transforms
cites work: On Fourier transforms of functions supported on sets of finite Lebesgue measure
cites work: The support of functions and distributions with a spectral gap.
cites work: Q5562324
cites work: Q5653133
cites work: Some Properties of the Fourier-Transform on Semisimple Lie Groups. III
cites work: On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups
cites work: Q5783325
cites work: Q4777532
cites work: Q5615626
cites work: Sur l'analyse harmonique du groupe affine de la droite
cites work: Q4068932
cites work: Q5683219
cites work: Spherical Functions Over -Adic Fields, II
cites work: Q4111790
cites work: Q3039597
cites work: Q3829996
cites work: Local Uncertainty Inequalities for Locally Compact Groups
description: scientific article; zbMATH DE number 4075768