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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Optimal logarithmic energy points on the unit sphere


Author: J. S. Brauchart
Journal: Math. Comp. 77 (2008), 1599-1613
MSC (2000): Primary 41A25; Secondary 31B15, 33C45, 70F10
DOI: https://doi.org/10.1090/S0025-5718-08-02085-1
Published electronically: February 6, 2008
MathSciNet review: 2398782
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Abstract: We study minimum energy point charges on the unit sphere $ \mathbb{S}^{d}$ in $ \mathbb{R}^{d+1}$, $ d\geq3$, that interact according to the logarithmic potential $ \log(1/r)$, where $ r$ is the Euclidean distance between points. Such optimal $ N$-point configurations are uniformly distributed as $ N\to\infty$. We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order $ \mathcal{O}(N^{-1/(d+2)})$. Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term $ (1/d)(\log N)/N$ in the asymptotical expansion of the optimal energy. Previously, this was known for the unit sphere in $ \mathbb{R}^{3}$ only. Furthermore, we present an upper bound for the error of integration for an equally-weighted numerical integration rule $ \numint_{N}$ with the $ N$ nodes forming an optimal logarithmic energy configuration. For polynomials $ p$ of degree at most $ n$ this bound is $ C_{d} ( N^{1/d} / n )^{-d/2} \Vert p\Vert _{\infty}$ as $ n/N^{1/d}\to0$. For continuous functions $ f$ of $ \mathbb{S}^{d}$ satisfying a Lipschitz condition with constant $ C_{f}$ the bound is $ (12dC_{f} + C_{d}^{\prime}\Vert f\Vert _{\infty}) \mathcal{O}(N^{-1/(d+2)})$ as $ N\to\infty$.


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Additional Information

J. S. Brauchart
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

DOI: https://doi.org/10.1090/S0025-5718-08-02085-1
Keywords: Discrepancy, Fekete points, logarithmic energy, Riesz energy, sphere, ultraspherical expansion
Received by editor(s): April 19, 2007
Received by editor(s) in revised form: June 17, 2007
Published electronically: February 6, 2008
Additional Notes: The research of this author was supported, in part, by the U. S. National Science Foundation under grant DMS-0532154 (D.P. Hardin and E.B. Saff principal investigators)
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.