Orthogonality of spherical harmonic coefficients

Abstract

An essentially arbitrary function V( θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics V(θ, Λ) = a ∑ n=1 ∞ ∑ m=0 n p m n (cos θ) (g m n cos mΛ + h m n sin mΛ) where the P n m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that 〈[P m n(cos θ) cos mΛ]〉 2 =1/ (2n+1) The angular brackets denote an average over the sphere. The class of functions V( θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g n m and h n m will be functions of the particular coordinate system orientation, but 〈:(g m n) 2〉) = 〈(h m n) 2 = S n/ (2n=1) where S n = ∑ m=0 n [(g m n) 2 + (h m n) 2] for any orientation of the coordinate system ( S n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈 g m n g l p 〉 = 〈 h m n h l p 〉 = 0 for l ≠ m or p ≠ n and 〈 g m n h l p 〉 = 0 fro all n, m, p, l.