An uncertainty principle for Hankel transforms

There exists a generalized Hankel transform of order α ≥ -1/2 on R, which is based on the eigenfunctions of the Dunkl operator \[ T_α f(x) = f^ (x) + (α +1/2 ) f(x)-f(-x)/x , f∈ C^1(R). \] For α =-1/2 this transform coincides with the usual Fourier transform on R. In this paper the operator T_α 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 α ≥ -1/2 on [0,∞ [.