Error in Sobolev norms of orthogonal projection onto polynomials in the unit ball

Leonardo E. Figueroa

Published 2016-07-04Version 1

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 generalized Gegenbauer form $x \mapsto (1-\|x\|^2)^\alpha$, $\alpha > -1$. Said properties are measured in Sobolev-type norms in which the same weighted $L^2$ norm is used to control all involved weak derivatives. The method of proof does not rely on any particular basis of orthogonal polynomials, which allows for a streamlined and dimension-independent exposition.