Geodesic
An uncertainty principle related to the Poisson summation formula
Studia Mathematica, Tome 121 (1996) no. 1, pp. 87-104

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

We prove a class of uncertainty principles of the form $∥S_{g}f∥_{1} ≤ C(∥x^{a}f∥_{p} + ∥ω^{b}f̂∥_{q})$, 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 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.
DOI : 10.4064/sm-121-1-87-104
Keywords: uncertainty principle, Poisson summation formula, unimodular polynomial, modulation space, time-frequency analysis, phase space

K. Gröchenig 1

1 
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K. Gröchenig. An uncertainty principle related to the Poisson summation formula. Studia Mathematica, Tome 121 (1996) no. 1, pp. 87-104. doi: 10.4064/sm-121-1-87-104

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