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

## Shayne Waldron

## Abstract:

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

## Availability: