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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) ℱα of order α>−½. Roughly speaking, if we denote by PWα(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 PWα(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.