On spherical expansions of zonal functions and Fourier transforms of SO(d)-finite measures

Abstract:
The article presents a simple new approach to the determination of the spherical expansion of zonal functions on euclidean space. We use the standard representation theoretic properties of spherical harmonics and the explicit form of the reproducing kernels for these spaces by means of classical Gegenbauer polynomials. As a special case we reobtain by this method the so called plane wave expansion and the expansion of the Poisson kernel. We present a new method of computing the Fourier transform of SO(d)-finite functions on the unit sphere which enables us to reobtain the classical Bochner identity.