Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight

Shayne Waldron


This paper considers tight frame decompositions of the Hilbert space P_n of orthogonal polynomials of degree n for a radially symmetric weight on R^d, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p in P_n with property that each f in P_n 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 p_\xi obtained from p by moving its pole to \xi in S:={\xi in R^d:|\xi|=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 P_n.

Keywords: multivariate orthogonal polynomials, Jacobi polynomials, Gegenbauer polynomials, ultraspherical polynomials, Legendre polynomials, Hermite polynomials, Laguerre polynomials, harmonic functions, spherical harmonics, zonal harmonics, quadrature for trigonometric polynomials, cubature on the sphere, representation theory, radial functions, ridge functions, zonal functions, tight frames

Math Review Classification: Primary 33C45, 33D50; Secondary 06B15, 42C15;

Length: 20 pages

Last Updated: 7 August 2007