Orthogonal polynomial projection error in Dunkl–Sobolev norms in the 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 reflection-invariant form (1−‖x‖2)α∏i=1d|xi|γi, α,γ1,…,γd>−1. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted L2 norm is used to control all the involved differential–difference Dunkl operators, such as those appearing in the Sturm–Liouville characterization of similarly weighted L2-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms.