Multivariate Jacobi polynomials with singular weights

First we give a compact treatment of the Jacobi polynomials on a simplex in IR 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–Bézier form. We then consider all aspects of the limiting case when the parameters μ = (μ0, . . . , μd) 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 μj → −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.