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

TL;DR

This paper investigates the approximation capabilities of weighted orthogonal projectors on the unit disk using Sobolev norms, employing Zernike polynomials to analyze error bounds and their sharpness.

Contribution

It provides new insights into Sobolev norm approximation errors for orthogonal projections with generalized Gegenbauer weights on the unit disk, including sharpness results and auxiliary properties.

Findings

Derived sharp error bounds for polynomial projections in Sobolev norms.

Established properties of Sobolev spaces with weighted norms on the disk.

Reported numerical tests support theoretical results.

Abstract

We study approximation properties of weighted $L^2$-orthogonal projectors onto the space of polynomials of degree less than or equal to $N$ on the unit disk where the weight is of the generalized Gegenbauer form $x \mapsto (1-|x|^2)^\alpha$. The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted $L^2$ norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases and otherwise strongly hinted at by reported numerical tests. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and non-uniformly weighted Sobolev spaces involved, a Markov-type inequality, connection coefficients between Zernike polynomials and relations between the Fourier-Zernike expansions of a function and its derivatives.