Vector-valued distributions and Hardy's uncertainty principle for operators
type: Item
instance of: scholarly article
title: Vector-valued distributions and Hardy's uncertainty principle for operators
zbMATH Open document ID: 1195.42050
DOI: 10.4171/RMI/597
published in: Revista Matemática Iberoamericana
publication date: 2010-05-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.1807
full work available at URL: https://projecteuclid.org/euclid.rmi/1266330120
review text: Let \(f\) be a function on \(\mathbb{R}^n\) such that \(\text{exp}(a |\cdot |^2) f\) and \(\text{exp}(b |\cdot |^2) \hat f\) are bounded, where \(a,b>0\) and \(\hat f\) denotes the Fourier transform of \(f\). According to \textit{G. H. Hardy}'s Uncertainty Principle [J. Lond. Math. Soc. 8, 227--231 (1933; Zbl 0007.30403; JFM 59.0425.01)], if \(ab>\pi^2\), then \(f=0\), while if \(ab=\pi^2\), then \(f=c\,\text{exp}(-a |\cdot |^2)\). The paper contains a survey of the most important publications related to this famous result. New interesting generalizations to vector-valued functions and operators are obtained.
reviewed by: Boris Rubin
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5708818
zbMATH Keywords: uncertainty principle
zbMATH Keywords: linear operators
zbMATH Keywords: Hardy's theorem
author: Michael G. Cowling
author: Bruno Demange
author: Maddala Sundari
MaRDI profile type: Publication
arXiv ID: 0805.1807
cites work: A survey on uncertainty principles related to quadratic forms
cites work: Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth
cites work: Q3436875
cites work: Q5653133
cites work: A Theorem Concerning Fourier Transforms
cites work: A Note on Fourier Transforms
cites work: An introduction to the uncertainty principle. Hardy's theorem on Lie groups. With a foreword by Gerald B. Folland
cites work: Q4002250
OpenAlex ID: W1977089840
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description: scientific article