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

The aim of this paper is to prove a quantitative extension of Shapiro’s result on the time–frequency concentration of orthonormal sequences in Lα2(R+). More precisely, we prove that, if {φn}n=0+∞ is an orthonormal sequence in Lα2(R+), then for all N≥0N≥0∑n=0N(‖xφn‖Lα22+‖ξHα(φn)‖Lα22)≥2(N+1)(N+1+α), and 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 Lα2(R+), that is if {φn}n=0+∞ is an orthonormal basis for Lα2(R+) and s>0s>0, then supn(‖xsφn‖Lα22‖ξsHα(φn)‖Lα22)=+∞.