The Logvinenko-Sereda Theorem for the Fourier-Bessel transform

Saifallah Ghobber (1, 2) , Philippe Jaming (3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Analyse harmonique et fonctions spéciales
3 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Fourier-Bessel transform; Hankel transform; uncertainty principle; strong annihilating pairs

Domaines: Analyse classique [math.CA]

Saifallah Ghobber, Philippe Jaming. The Logvinenko-Sereda Theorem for the Fourier-Bessel transform. Integral Transforms and Special Functions, 2013, 24, pp.470-484. ⟨hal-00696009⟩