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

TL;DR

This paper investigates the approximation capabilities of weighted orthogonal projectors in Sobolev norms within the unit ball, providing a basis-independent analysis that enhances understanding of polynomial approximation in weighted Sobolev spaces.

Contribution

It introduces a basis-independent method to analyze polynomial projection errors in Sobolev norms on the unit ball, broadening the theoretical framework for approximation theory.

Findings

Error estimates for weighted orthogonal projections in Sobolev norms

Dimension-independent analysis of approximation properties

Method applicable without specific polynomial bases

Abstract

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 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.