An uncertainty principle related to the Poisson summation formula

K. Gröchenig

Studia Mathematica (1996)

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

Abstract

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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 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.

How to cite

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Gröchenig, 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 -

References

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