# An uncertainty principle related to the Poisson summation formula

Studia Mathematica (1996)

- Volume: 121, Issue: 1, page 87-104
- ISSN: 0039-3223

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topGröchenig, K.. "An uncertainty principle related to the Poisson summation formula." Studia Mathematica 121.1 (1996): 87-104. <http://eudml.org/doc/216344>.

@article{Gröchenig1996,

abstract = {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.},

author = {Gröchenig, K.},

journal = {Studia Mathematica},

keywords = {uncertainty principle; Poisson summation formula; unimodular polynomial; modulation space; time-frequency analysis; phase space; uncertainty principles; short time Fourier transform; Gabor expansions; unimodular polynomials},

language = {eng},

number = {1},

pages = {87-104},

title = {An uncertainty principle related to the Poisson summation formula},

url = {http://eudml.org/doc/216344},

volume = {121},

year = {1996},

}

TY - JOUR

AU - Gröchenig, K.

TI - An uncertainty principle related to the Poisson summation formula

JO - Studia Mathematica

PY - 1996

VL - 121

IS - 1

SP - 87

EP - 104

AB - 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.

LA - eng

KW - uncertainty principle; Poisson summation formula; unimodular polynomial; modulation space; time-frequency analysis; phase space; uncertainty principles; short time Fourier transform; Gabor expansions; unimodular polynomials

UR - http://eudml.org/doc/216344

ER -

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