A simple Proof of Stolarsky's Invariance Principle

TL;DR

This paper provides a straightforward proof of Stolarsky's invariance principle, connecting point distributions on spheres, discrepancy measures, and potential theory through reproducing kernel Hilbert spaces.

Contribution

It offers a simple, reproducing kernel Hilbert space-based proof of Stolarsky's invariance principle, clarifying its theoretical foundations.

Findings

Simplifies the proof of Stolarsky's invariance principle

Links discrepancy and potential theory on spheres

Uses reproducing kernel Hilbert spaces for the proof

Abstract

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the uniform measure on the sphere a potential-theoretical quantity (Bj{\"o}rck [Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa (first author and Womersley [Preprint]). In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.