## 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