Analogues of Beurling's theorem for some integral transforms

Abstract

A famous result of Beurling says that if the Fourier transform of a non-zero integrable function on the real line has certain exponential decay, then the function cannot vanish on a set of positive Lebesgue measure. We prove an analogue of Beurling's theorem for Hankel transform and some several variable analogues of Beurling's theorem for Fourier transform as a consequence of similar results for Dunkl transform. We also prove an analogue of Beurling's theorem for spectral projections associated to the Dunkl-Laplacian.