The Fourier Transform Associated to the k-Hyperbolic Dirac Operator

Abstract

The polynomial null solutions of the k-hyperbolic Dirac operator are investigated by the \(\mathfrak {osp}(1|2)\) approach. These solutions are then utilized to construct the (fractional) Fourier transform associated to the k-hyperbolic Dirac operator. The resulting integral kernels are found to be a specific kind of Dunkl kernels. Additionally, we give tight uncertainty inequalities for three distinct fractional Fourier transforms that we have defined. These inequalities are new even for the ordinary fractional Hankel and Weinstein transforms.