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arXiv · 1101.4448

A simple Proof of Stolarsky's Invariance Principle

Johann S. Brauchart·Josef Dick
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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.

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Johann S. Brauchart, Josef Dick. 2011-01-24. A simple Proof of Stolarsky's Invariance Principle. https://doi.org/10.1090/s0002-9939-2013-11490-5

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