Orthogonal Bases for Spaces of Complex Spherical Harmonics
نویسندگان
چکیده
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 orthogonality with respect to the inner product of L(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 differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L(Ω2q) inner product. Bi-orthogonality in (P(Cq), [·, ·]) is fully investigated.
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