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An uncertainty principle related to the Poisson summation formula

K. Gröchenig — 1996

Studia Mathematica

We prove a class of uncertainty principles of the form S g f 1 C ( x a f p + ω b f ̂ q ) , where S g f 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...

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