Sobolev orthogonal polynomials defined via gradient on the unit ball

Abstract

An explicit family of polynomials on the unit ball $B^d$ of $\RR^d$ is constructed, so that it is an orthonormal family with respect to the inner product $$ < f,g > = ρ\int_{B^d}\nabla f(x)\cdot \nabla g(x) dx + \CL (fg), $$ where $ρ>0$, $\nabla$ is the gradient, and $\CL(fg)$ is either the inner product on the sphere $S^{d-1}$ or $f(0)g(0)$.