A family of Sobolev Orthogonal Polynomials on the Unit Ball

A family of orthonormal polynomials on the unit ball $B^d$ of \RR^d with respect to the inner product $$= \int_{B^d}\Delta[(1-\|x\|^2) f(x)] \Delta[(1-\|x\|) g(x)] dx,$$ where $\Delta$ is the Laplace operator, is constructed explicitly

Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced

Sobolev orthogonal polynomials on a simplex

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