Abstract
We give an extension of Stolarsky’s invariance principle. For a certain class of energy kernel functions we show relations between the spherical cap discrepancy and the energy of N—point sets on the unit sphere. We see that cap discrepancy and energy are comparable in a sense. In addition an integral representation for the energy which involves discrepancy functions is derived.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Abramowitz, I. A. Stegun (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1970.
G. Andrews, R. Askey, R. Roy: Special Functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, 1999.
J. S. Brauchart: Punktverteilungen extremaler diskreter Energie auf Sphären. Diploma thesis, Technical University Graz, 2001.
P. J. Grabner, R. F. Tichy: Spherical designs, discrepancy and numerical integration, Math. Comp. 60 (1993), no. 201, 327–336.
I. J. Schoenberg: Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108.
K. B. Stolarsky: Sums of distances between points on a sphere II, Proc. Amer. Math. Soc. 41 (1973), 575–582.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this paper
Cite this paper
Brauchart, J.S. (2003). Note on a Generalized Invariance Principle and its Relevance for Cap Discrepancy and Energy. In: Haussmann, W., Jetter, K., Reimer, M., Stöckler, J. (eds) Modern Developments in Multivariate Approximation. International Series of Numerical Mathematics, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8067-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8067-1_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9427-2
Online ISBN: 978-3-0348-8067-1
eBook Packages: Springer Book Archive