ArticleOriginal scientific text
Title
An uncertainty principle related to the Poisson summation formula
Authors 1
Affiliations
- Department of Mathematics U-9, The University of Connecticut, Storrs, Connecticut 06269-3009, U.S.A.
Abstract
We prove a class of uncertainty principles of the form
,
where is the short time Fourier transform of f. We obtain a characterization of the range of parameters a,b,p,q for which such an uncertainty principle holds. Counter-examples are constructed using Gabor expansions and unimodular polynomials. These uncertainty principles relate the decay of f and f̂ to their behaviour in phase space. Two applications are given: (a) If such an inequality holds, then the Poisson summation formula is valid with absolute convergence of both sums. (b) The validity of an uncertainty principle implies sufficient conditions on a symbol σ such that the corresponding pseudodifferential operator is of trace class.
Keywords
uncertainty principle, Poisson summation formula, unimodular polynomial, modulation space, time-frequency analysis, phase space
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