The Logvinenko-Sereda Theorem for the Fourier-Bessel transform

The aim of this paper is to establish an analogue of Logvinenko-Sereda’s theorem for the Fourier-Bessel transform (or Hankel transform) ${\text{ff}}_{\alpha }$ of order $\alpha >-1\mathrm{/}2$. Roughly speaking, if we denote by $P{W}_{\alpha }(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction map $f\to f{\mathrm{\mid }}_{\mathrm{\Omega }}$ is essentially invertible on $P{W}_{\alpha }(b)$ if and only if $\mathrm{\Omega }$ 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.