Real Paley-Wiener theorems for the (k,a)-generalized Fourier transform

摘要

In this paper, we obtain several versions of the real Paley-Wiener theorems for the (k,a)-generalized Fourier transform recently investigated at length by Ben Said, Kobayashi, and orsted. This generalized Fourier transform can be regarded as a two-parameter generalization of Howe's description of classical Fourier transform, where k is a multiplicity function for the Dunkl operators on Rd set minus {0} and a>0 arises from the interpolation of the two Lie algebra sl(2,R) actions on the Weil representation of Mp(d,R) and the minimal unitary representation of the O(d+1,2).