A simple proof of Stolarsky's invariance principle
Authors:
Johann S. Brauchart and Josef Dick
Journal:
Proc. Amer. Math. Soc. 141 (2013), 20852096
MSC (2010):
Primary 41A30; Secondary 11K38, 41A55
Published electronically:
January 29, 2013
MathSciNet review:
3034434
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Additional Information
Abstract: Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap discrepancy to give the distance integral of the uniform measure on the sphere which is a potentialtheoretical quantity (Björck [Ark. Mat. 3 (1956), 255269]). Read differently it expresses the worstcase numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the 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/S000299392013114905
Keywords:
Invariance principle,
reproducing kernel Hilbert space,
sphere,
spherical cap discrepancy,
sum of distances,
worstcase 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 APARTFellowship 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.
