A simple Proof of Stolarsky's Invariance Principle
Numerical Analysis
2014-02-17 v1
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 -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 -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.
Keywords
Cite
@article{arxiv.1101.4448,
title = {A simple Proof of Stolarsky's Invariance Principle},
author = {Johann S. Brauchart and Josef Dick},
journal= {arXiv preprint arXiv:1101.4448},
year = {2014}
}