Mathematics – Classical Analysis and ODEs
Scientific paper
2006-06-16
Journal of Functional Analysis 243 (15/02/2007) 611-630
Mathematics
Classical Analysis and ODEs
Scientific paper
10.1016/j.jfa.2006.09.001
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 sphero\"{i}dal 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.
Jaming Philippe
Powell Alexander M.
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