Geodesic cycles on the Sphere: $t$-designs and Marcinkiewicz-Zygmund Inequalities

TL;DR

This paper investigates geodesic cycles on the sphere, focusing on their application in creating equal-weight quadrature rules and approximation methods, highlighting their mathematical properties and practical relevance.

Contribution

It introduces a new perspective on geodesic cycles' role in quadrature rules and approximation on the sphere, linking geometric structures with analytical inequalities.

Findings

Characterization of geodesic cycles on the sphere.

Development of Marcinkiewicz-Zygmund inequalities for these cycles.

Applications to quadrature and approximation methods.

Abstract

A geodesic cycle is a closed curve that connects finitely many points along geodesics. We study geodesic cycles on the sphere in regard to their role in equal-weight quadrature rules and approximation.