Sobolev Orthogonal Polynomials on a Simplex

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_{\gamma}(x)=x^{\gamma_1}_1\ \cdot\cdot\cdot X^{\gamma d}_d(1-|x|)^{\gamma d+1}$ when all γi>−1 and they are eigenfunctions of a second-order partial differential operator Lγ. The singular cases that some, or all, γ1,…,γd+1 are −1 are studied in this paper. First, a complete basis of polynomials that are eigenfunctions of Lγ 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.