Uncertainty principles for orthonormal sequences |
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Authors: | Philippe Jaming Alexander M Powell |
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Institution: | a Université d'Orléans, Faculté des Sciences, MAPMO - Fédération Denis Poisson, BP 6759, F 45067 Orléans cedex 2, France b Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA |
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Abstract: | The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L2(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L2(R) then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro's uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given. |
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Keywords: | Uncertainty principle Spherical code Orthonormal basis Hermite functions Prolate spheroï dal wave functions Riesz basis |
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