A simple proof of Stolarsky’s invariance principle

Brauchart, Johann; Dick, Josef

doi:10.1090/S0002-9939-2013-11490-5

A simple proof of Stolarsky’s invariance principle
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by Johann S. Brauchart and Josef Dick

Proc. Amer. Math. Soc. 141 (2013), 2085-2096

DOI: https://doi.org/10.1090/S0002-9939-2013-11490-5

Published electronically: January 29, 2013

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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 which is a potential-theoretical quantity (Björck [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. In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.

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