AN UNCERTAINTY PRINCIPLE FOR HANKEL TRANSFORMS

There exists a generalized Hankel transform of order af > -1/2 on R, which is based on the eigenfunctions of the Dunkl operator Tc, f (x) = f , (X) + (ce + 1 ) f (x)-f (-x) f EE C1 (R). Taf(X) = ~~2 x For a = -1/2 this transform coincides with the usual Fourier transform on R. In this paper the operator Tog replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on R. It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on L2 (R); moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order a > -1/2 on [Ooc40.

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