# Hermite polynomials on the plane

## Abstract:

The space $\cP_n$ of bivariate generalised Hermite polynomials of degree $n$ is invariant under rotations. We exploit this symmetry to construct an orthonormal basis for $\cP_n$ which consists of the rotations of a single polynomial through the angles ${\ell\pi\over n+1}$, $\ell=0,\ldots n$. Thus we obtain an orthogonal expansion which retains as much of the symmetry of $\cP_n$ as is possible. Indeed we show that a continuous version of this orthogonal expansion exists.

Keywords: bivariate Hermite polynomials, Laguerre polynomials, Zernike polynomials, quadrature for trigonometric polynomials, representation theory, orthogonal expansions

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

Length: 7 pages

Last Updated: 19 March 2007