Orthogonal Bases for Spaces of Complex Spherical Harmonics

Abstract

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2 q of . The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L 2 (Ω 2 q ). As a bypass, we study the inner product over the space of polynomials in the variables z , , in which ƒ( D ̅) is the differential operator with symbol ƒ( z ̄). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L 2 (Ω 2 q ) inner product. Bi-orthogonality in (, [·, ·]) is fully investigated.