An uncertainty principle for eigenfunction expansions.
type: Item
instance of: scholarly article
title: An uncertainty principle for eigenfunction expansions.
zbMATH Open document ID: 1056.42511
author: S. Thangavelu
author: Vishwambhar Pati
author: Alladi Sitaram
author: Maddala Sundari
published in: The Journal of Fourier Analysis and Applications
publication date: 1998-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/59489
review text: The aim of this paper is to present evidence that there are uncertainty principles associated to eigenfunction expansions. Several examples are given. If \(f\) is a radial function on a hyperboloid such that \(|f(x)| \leq C \exp(\alpha d (x,o)^2)\), \(o\) being a fixed reference point, and if the spherical transform \(\widetilde{f}\) of \(f\) satisfies \(|\widetilde{f} (\lambda)| \leq C \exp(-\beta |\lambda|^2)\), where \(\alpha \beta >1/4\), then \(f=0\). If \(f\) is a function on a compact connected analytic manifold \(M\), written as \(\sum f_k\), where the \(f_k\) are all eigenfunctions of a fixed analytic elliptic operator on \(M\), and if the \(f_k\) decay fast enough, then \(f\) must be analytic, and this cannot be localised in \(M\). Finally, a similar result is proved for Hermite expansions of functions on \(R^n\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1153119
reviewed by: Michael G. Cowling
MaRDI profile type: Publication
OpenAlex ID: W2060654180
DOI: 10.1007/S00041-001-4036-2
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description: scientific article; zbMATH DE number 1153119