On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

We prove that, if the coefficients of a Fourier--Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO(3) and the principal series of the unitary representation of SU(1,1)$.