The Logvinenko-Sereda Theorem for the Fourier-Bessel transform

TL;DR

This paper extends the Logvinenko-Sereda theorem to the Fourier-Bessel transform, characterizing when functions can be reconstructed from their values on dense sets and providing inverse norm estimates.

Contribution

It establishes an analogue of the Logvinenko-Sereda theorem for the Fourier-Bessel transform and offers a Bernstein type inequality for this transform.

Findings

Restriction map invertibility on dense sets characterized

Norm estimates for inverse map provided

Bernstein inequality for Fourier-Bessel transform proved

Abstract

The aim of this paper is to establish an analogue of Logvinenko-Sereda's theorem for the Fourier-Bessel transform (or Hankel transform) $\ff_\alpha$ of order $\alpha>-1/2$. Roughly speaking, if we denote by $PW_\alpha(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction map $f\to f|_\Omega$ is essentially invertible on $PW_\alpha(b)$ if and only if $\Omega$ is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result we prove a Bernstein type inequality for the Fourier-Bessel transform.