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) acting 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 introduced 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 functions 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) and the Fourier-Legendre expansion of invariant kernels on the sphere Sd-1.