Sobolev orthogonal polynomials on a simplex

Abstract

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1- |x|)^{\g_{d+1}} when all \g_i > -1 and they are eigenfunctions of a second order partial differential operator L_\bg. The singular cases that some, or all, \g_1,...,\g_{d+1} are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of L_\bg in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.Comment: 32 page