The aim of this paper is to establish an analogue of Logvinenko–Sereda's theorem for the Fourier–Bessel transform (or Hankel transform) ℱ_α of order α>−½. Roughly speaking, if we denote by PW_α(b) the Paley–Wiener space of L ²-functions with the Fourier–Bessel transform supported in [0, b], then we show that the restriction map f→f|_Ω is essentially invertible on PW_α(b) if and only if Ω 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.

The aim of this paper is to prove new uncertainty principles for an integral operator with a bounded kernel. To do so we prove a Nash-type inequality and a Carlson-type inequality for this transformation. From this we deduce a variation on Heisenberg’s uncertainty inequality and Faris’s local uncertainty principle. We also prove a variation on Donoho-Stark’s uncertainty principle. Our results can be applied to a wide variety of integral operators, including the Fourier transform, the Fourier-Bessel transform, the generalized Fourier transform and the -transform.