CONNECTION BETWEEN THE HARMONIC-ANALYSIS ON THE SPHERE AND THE HARMONIC-ANALYSIS ON THE ONE-SHEETED HYPERBOLOID - AN ANALYTIC CONTINUATION VIEWPOINT .3.

A Fourier-Laplace transformation L(d) (d greater than or equal to 3) a cting on a class of holomorphic functions on the complex quadric X(d-1 )((c)) with equation z((0)2) - z((1)2) - ... -z((d-1)2) = -1 is introd uced and studied. Its expression as the composition product L(d) = L o R(d)((c)) makes use of the complex Radon-Abel transformation R(d)((c) ) on X(d-1)((c)), studied in Part II and of a one dimensional Fourier- Laplace transformation L acting on relevant subspaces of holomorphic f unctions introduced in Part I. This transformation L(d) allows one to relate by analytic continuation the (''spherical'') Laplace transform of invariant Volterra kernels on the one-sheeted hyperboloid X(d-1) an d the Fourier-Legendre expansion of invariant kernels on the sphere Sd -1.