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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A simple proof of Stolarsky’s invariance principle
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by Johann S. Brauchart and Josef Dick PDF
Proc. Amer. Math. Soc. 141 (2013), 2085-2096 Request permission

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.
References
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Additional Information
  • Johann S. Brauchart
  • Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
  • MR Author ID: 730033
  • Email: j.brauchart@unsw.edu.au
  • Josef Dick
  • Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
  • Email: josef.dick@unsw.edu.au
  • Received by editor(s): January 23, 2011
  • Received by editor(s) in revised form: October 5, 2011
  • Published electronically: January 29, 2013
  • Additional Notes: The first author was supported by an APART-Fellowship of the Austrian Academy of Sciences.
    The second author was supported by an Australian Research Council Queen Elizabeth II Fellowship.
  • Communicated by: Walter Van Assche
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2085-2096
  • MSC (2010): Primary 41A30; Secondary 11K38, 41A55
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11490-5
  • MathSciNet review: 3034434