A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Abstract This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential ei(x|y) with x, y ∈ Rd . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components and avoids using any integral identities. The proof makes also use of 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. In the last section of the paper a new method of computing the Fourier transforms of SO(d)-finite functions on the unit sphere is presented which enables us to reobtain both the classical Bochner identity and generalizations of it due to one of the present authors and F. J. Gonzalez Vieli.