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Proceedings of the American Mathematical Society
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A simple proof of Stolarsky's invariance principle


Authors: Johann S. Brauchart and Josef Dick
Journal: Proc. Amer. Math. Soc. 141 (2013), 2085-2096
MSC (2010): Primary 41A30; Secondary 11K38, 41A55
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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Additional Information

Johann S. Brauchart
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11490-5
PII: S 0002-9939(2013)11490-5
Keywords: Invariance principle, reproducing kernel Hilbert space, sphere, spherical cap discrepancy, sum of distances, worst-case numerical integration error
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.