Uncertainty principles on certain Lie groups
type: Item
instance of: scholarly article
title: Uncertainty principles on certain Lie groups
zbMATH Open document ID: 0857.43011
DOI: 10.1007/BF02880360
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1997-03-03T00:00:00Z
review text: The authors show how various uncertainty principles can be formulated in the case of some locally compact groups including \(\mathbb{R}^n\), the Heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane. A theorem of Hardy, which is one way of formulating the uncertainty principle for the Fourier transform \(\widehat f\), is the following Theorem (Hardy): Suppose \(f\) is a measurable function on \(\mathbb{R}\) such that (1) \(|f(x)|\leq Ce^{-\alpha x^2}\), and \(|\widehat f(\xi) |\leq Ce^{-\beta \xi^2}\), \(x,\xi\in \mathbb{R}\), \(\alpha, \beta>0\). If \(\alpha\beta > {1\over 4}\) then \(f=0\) a.e. If \(\alpha \beta< {1\over 4}\) then there are infinitely many linearly independent functions satisfying (1) and if \(\alpha\beta = {1\over 4}\) then \(f(x)=Ce^{-\alpha x^2}\). -- The authors prove an analogue of Hardy's theorem for \(\mathbb{R}^n\) \((n\geq 2)\), the Heisenberg group and the Euclidean motion group of the plane, and show that though the exact analogue for the reduced Heisenberg group fails, a slightly modified version continues to hold. Furthermore, they show another way of formulating the uncertainty principle for the Fourier transform on the Heisenberg group in terms of the supports of the function \(f\) and its Fourier transform \(\widehat f\). They formulate and prove a local uncertainty inequality. From this uncertainty inequality they deduce a global inequality similar to the classical Heisenberg-Pauli-Weyl uncertainty inequality.
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 903346
zbMATH Keywords: uncertainty principles
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Euclidean motion group
zbMATH Keywords: Fourier transform
zbMATH Keywords: Hardy's theorem
zbMATH Keywords: Heisenberg-Pauli-Weyl uncertainty inequality
Wikidata QID: Q115391354
author: Alladi Sitaram
author: Maddala Sundari
author: S. Thangavelu
reviewed by: Koichi Saka
MaRDI profile type: Publication
cites work: On Fourier transforms of functions supported on sets of finite Lebesgue measure
cites work: Q3829996
cites work: Q5653133
cites work: A Qualitative Uncertainty Principle for Certain Locally Compact Groups
cites work: Q4089199
cites work: Harmonic Analysis in Phase Space. (AM-122)
cites work: Spherical Harmonics, the Weyl Transform and the Fourier Transform on the Heisenberg Group
cites work: The Radon transform
cites work: A Qualitative Uncertainty Principle for Unimodular Groups of Type I
cites work: A uniqueness theorem of Beurling for Fourier transform pairs
cites work: Q3706198
cites work: Functions and their Fourier transforms with supports of finite measure for certain locally compact groups
cites work: Local Uncertainty Inequalities for Locally Compact Groups
cites work: Q4111790
cites work: Q3742109
cites work: Some uncertainty inequalities
cites work: Some restriction theorems for the Heisenberg group
cites work: Q3137751
full work available at URL: https://doi.org/10.1007/bf02880360
OpenAlex ID: W2012012552
description: scientific article; zbMATH DE number 903346