Multivariate Jacobi polynomials with singular weights

Shayne Waldron


First we give a compact treatment of the Jacobi polynomials on a simplex in $\R^d$ which exploits and emphasizes the symmetries that exist. This includes the various ways that they can be defined: via orthogonality conditions, as a hypergeometric series, as eigenfunctions of an elliptic pde, as eigenfunctions of a positive linear operator, and through conditions on the Bernstein--B\'ezier form. We then consider all aspects of the limiting case when the parameters $\mu=(\mu_0,\ldots,\mu_d)$ of the Jacobi polynomials approach $-1$, and the weight becomes singular. We show that the orthogonal projection of a continuous function onto the Jacobi polynomials of degree $n$ has a limit as the $\mu_j\to-1$, and give an explicit formula for the corresponding `orthogonal' expansion. It turns out that this expansion is closely related to the limit of the eigenfunction expansion of the Bernstein operator and a new mean value interpolant.

Keywords: Jacobi polynomials on a simplex, mean value interpolation, multivariate Bernstein operator, singular weight function, tight frame,

Math Review Classification: Primary 33C45, 41A10, 41A05, 41A63; Secondary 15A18, 42C15, 30E05

Length: 16 pages

Last Updated: 13 February 2007