Orthogonal polynomials of several variables for a radially symmetric weight

This paper considers tight frame decompositions of the Hilbert space Pn of orthogonal polynomials of degree n for a radially symmetric weight on IR, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p ∈ Pn with property that each f ∈ Pn can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials pξ obtained from p by moving its pole to ξ ∈ S := {ξ ∈ IR : |ξ| = 1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for Pn.