Orthogonal polynomial projection error in Dunkl-Sobolev norms in the ball

Abstract

We study approximation properties of weighted ${\mathrm{L}}^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form $(1-\Vert x{\Vert }^2{)}^{\alpha }{\prod }_{i=1}^d|x_i{|}^{{\gamma }_i}$, $\alpha ,{\gamma }_1,\dots ,{\gamma }_d>-1$. Said properties are measured in Dunkl-Sobolev-type norms in which the same weighted ${\mathrm{L}}^2$ norm is used to control all the involved differential-difference Dunkl operators, such as those appearing in the Sturm-Liouville characterization of similarly weighted ${\mathrm{L}}^2$-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.

arXiv comment27 pages

References in corpus (2)

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