Gonzalo A. Benavides, Leonardo E. Figueroa:
Orthogonal polynomial projection error in Dunkl-Sobolev norms in the ball
We study approximation properties of weighted L^2-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)^alpha |x_1|^(gamma_1) * ... * |x_d|^(gamma_d), alpha, gamma_1, ..., gamma_d > -1. Said properties are measured in Dunkl-Sobolev-type norms in which the same weighted L^2 norm is used to control all the involved differential-difference Dunkl operators, such as those appearing in the Sturm-Liouville characterization of similarly weighted L^2-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms. The method of proof relies on spaces instead of bases of orthogonal polynomials, which greatly simplifies the exposition.