On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases

TL;DR

This paper investigates the numerical computation of Marcinkiewicz-Zygmund constants for cubature rules and examines their impact on polynomial approximation using orthogonal bases across various domains, comparing different approximation methods.

Contribution

It introduces a numerical approach to compute Marcinkiewicz-Zygmund constants and analyzes their influence on polynomial approximation methods on multiple geometric domains.

Findings

Marcinkiewicz-Zygmund constants are crucial for understanding approximation quality.

Least squares projection performance is comparable to hyperinterpolation methods.

Open-source Matlab codes facilitate reproducibility and further research.

Abstract

We compute numerically the $L^2$ Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test some relevant rules on domains such as the interval, the square, the disk, the triangle, the cube and the sphere. The approximation power of the corresponding least squares (LS) projection is compared with standard hyperinterpolation and its recently proposed ``exactness-relaxed'' version. The Matlab codes used for these tests are available in open-source form.