# Multivariate Jacobi polynomials with singular weights

## Shayne Waldron

## Abstract:

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

## Availability: