A simple proof of Stolarsky's invariance principle

Johann Brauchart, Josef Dick

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)2085-2096
JournalProceedings of the American Mathematical Society
Volume141
Issue number6
DOIs
Publication statusPublished - 2013

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Basic - Fundamental (Grundlagenforschung)
  • Application
  • Theoretical

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