Sobolev orthogonal polynomials on a simplex

TL;DR

This paper investigates Sobolev orthogonal polynomials on a simplex, focusing on singular cases where parameters are -1, and constructs explicit orthogonal bases involving derivatives.

Contribution

It provides a complete basis of eigenpolynomials and explicitly determines the Sobolev inner product in singular cases.

Findings

Constructed eigenpolynomial bases for singular parameter cases

Explicit Sobolev inner product involving derivatives

Extended orthogonality results to singular parameter regimes

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.