Recall that a function is *harmonic* if its Laplacian is zero. Let $mathrm{Harm}(n,k)$ denote the vector space of $n$-variate harmonic polynomials that are homogeneous with degree $k$. When working with spherical harmonics, we endow this vector space with the inner product $langlecdot,cdotrangle$ defined in terms of the uniform probability measure over the unit sphere $mathbb{S}^{n-1}$. Many proofs involving spherical harmonics pass to an implicit orthogonal basis for this inner product space, but for computations, it is sometimes helpful to have an explicit basis.

Question.Is there a “nice” choice of orthogonal basis for $(mathrm{Harm}(n,k),langlecdot,cdotrangle)$? In particular, is

there a choice for which there exists a fast algorithm to compute an

arbitrary decomposition in the basis (à la FFT)?