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 Ω2q of Cq. The bases are shown to have many interesting properties, among them or-thogonality with respect to the inner product of L2(Ω2q). As a bypass, we study the inner product [f, g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables z, z ∈ Cq, in which f(D) is the differen-tial operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L2(Ω2q) inner product. Bi-orthogonality in (P(Cq), [·, ·]) is fully investigated. 1