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1996 | 121 | 1 | 87-104
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An uncertainty principle related to the Poisson summation formula

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Abstrakty
EN
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.
Twórcy
  • Department of Mathematics U-9, The University of Connecticut, Storrs, Connecticut 06269-3009, U.S.A., groch@math.uconn.edu
Bibliografia
  • [1] J. J. Benedetto, Frame decompositions, sampling, and uncertainty principle inequalities, in: Wavelets: Mathematics and Applications, J. Benedetto and M. Frazier (eds.), CRC Press, Boca Raton, 1994, 247-304.
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  • [8] H. G. Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Proc. Conf. "Constructive Function Theory", Edmonton, July 1986, Rocky Mountain J. Math. 19 (1989), 113-126.
  • [9] H. G. Feichtinger, personal communication.
  • [10] H. Feichtinger and K. Gröchenig, Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view, in: Wavelets: A Tutorial in Theory and Applications, Ch. K. Chui (ed.), Academic Press, Boston, 1992, 359-397.
  • [11] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, 1989.
  • [12] K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1-42.
  • [13] C. Heil, J. Ramanathan and P. Topiwala, Singular values of compact pseudodifferential operators, preprint, 1995.
  • [14] C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), 628-666.
  • [15] J.-P. Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc. 12 (1980), 321-342.
  • [16] J.-P. Kahane et P. G. Lemarié-Rieusset, Remarques sur la formule sommatoire de Poisson, Studia Math. 109 (1994), 303-316.
  • [17] Y. Katznelson, Une remarque concernant la formule de Poisson, ibid. 29 (1967), 107-108.
  • [18] E. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (3) (1990), 594-599.
  • [19] V. Losert, A characterization of the minimal strongly character invariant Segal algebra, Ann. Inst. Fourier (Grenoble) 30 (1980), 129-139.
  • [20] J. F. Price, Inequalities and local uncertainty principles, J. Math. Phys. 24 (1983), 1711-1714.
  • [21] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv121i1p87bwm
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