2006/06/16 by Philippe Jaming, Alexander M. Powell
Computer Science · Mathematics · #Bounded function #Combinatorics #Digital Filter Design and Implementation #Fourier series #Fourier transform #Function (biology) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Orthonormal basis #Pointwise #Pure mathematics #Sequence (biology) #Uniform boundedness #math.CA #msc:42B10
paper · pdf · doi:10.1016/j.jfa.2006.09.001
published as Journal of Functional Analysis 243 (15/02/2007) 611-630
arxiv created 2006/06/16 · openalex publication_date 2006/10/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.