Orthogonal polynomial projection error measured in Sobolev norms in the unit ball

Abstract

We study approximation properties of weighted L2-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the generalized Gegenbauer form x↦(1−‖x‖2)α, α>−1. Said properties are measured in Sobolev-type norms in which the same weighted L2 norm is used to control all the involved weak derivatives. The method of proof does not rely on any particular basis of orthogonal polynomials, which allows for a short, streamlined and dimension-independent exposition.