Band-limited functions: 𝐿^{𝑝}-convergence

Abstract

We consider the set B p ( Ω ) {B_p}(\Omega ) (functions of L p ( R ) {L^p}({\mathbf {R}}) whose Fourier spectrum lies in [ − Ω , + Ω ] [ - \Omega , + \Omega ] ). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if 4 / 3 > p > 4 4/3 > p > 4 . The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.