Phase space localization of orthonormal sequences in Lα2(R+)

Abstract

The aim of this paper is to prove a quantitative extension of Shapiro's result on the time-frequency concentration of orthonormal sequences in L 2 α (R + ). More precisely, we prove that, if {ϕ n } +∞ n=0 is an orthonormal sequence in L 2 α (R + ), then for all N ≥ 0and the equality is attained for the sequence of Laguerre functions. Particularly if the elements of an orthonormal sequence and their Fourier-Bessel transforms (or Hankel transforms) have uniformly bounded dispersions then the sequence is finite. Moreover we prove the following strong uncertainty principle for bases for L 2 α (R + ), that is if {ϕ n } +∞ n=0 is an orthonormal basis for L 2 α (R + ) and s > 0, then

Shapiro’s Uncertainty Principle in the Dunkl setting

“…We refer the reader to [1,2,8,11,16] for numerous results and discussions on timefrequency localisation of orthonormal sequences and bases.…”

Logarithmic uncertainty principles for the Hankel wavelet transform

“…Uncertainty principles in Fourier analysis set a limit to the possible concentration of a function and its Hankel transform in the time-frequency domain. The most familiar form is the Heisenbergtype inequalities, in which concentration is measured by dispersions (see [4,19,8]…”

Fourier-Like Multipliers and Applications for Integral Operators

“…[20]), the Hankel transform (see e.g. [13]), the the Dunkl transform (see e.g. [14]), the G-transform (see e.g.…”

“…Moreover the proof of Theorem A is inspired from the classical result for the Fourier transform in [20], where the author prove that Inequality (1.11) is sharp and the equality cases are attained for the sequence of Hermite functions, (see also [19]). For the Hankel transform [13] this inequality is also optimal and the optimizers are the sequence of Laguerre functions.…”