Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere

TL;DR

This paper demonstrates that certain point configurations on the 2-sphere serve as optimal polynomial meshes and introduces Caratheodory-Tchakaloff submeshes for efficient polynomial approximation.

Contribution

It provides an elementary proof linking covering point configurations to optimal polynomial meshes and constructs CATCH submeshes for compressed least squares fitting.

Findings

Good covering point configurations are optimal polynomial meshes on the 2-sphere.

Caratheodory-Tchakaloff submeshes enable efficient polynomial approximation.

The approach simplifies the understanding of polynomial meshes on the sphere.

Abstract

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting.