Uncertainty principles for orthonormal sequences

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 $L^2 (\R)$. 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 $L^2(\R)$ 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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Journal of Functional Analysis, Elsevier, 2007, 243, pp.611-630. 〈10.1016/j.jfa.2006.09.001〉
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Soumis le : vendredi 16 juin 2006 - 17:44:19
Dernière modification le : jeudi 3 mai 2018 - 15:32:06
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Philippe Jaming, Alexander Powell. Uncertainty principles for orthonormal sequences. Journal of Functional Analysis, Elsevier, 2007, 243, pp.611-630. 〈10.1016/j.jfa.2006.09.001〉. 〈hal-00080455〉

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