A family of Sobolev Orthogonal Polynomials on the Unit Ball

Abstract

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