Optimal logarithmic energy points on the unit sphere
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Abstract:
We study minimum energy point charges on the unit sphere $\mathbb {S}^{d}$ in $\mathbb {R}^{d+1}$, $d\geq 3$, 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 $Q_{N}$ with the $N$ nodes forming an optimal logarithmic energy configuration. For polynomials $p$ of degree at most $n$ this bound is $C_{d} ( N^{1/d} / n )^{-d/2} \| p\|_{\infty }$ as $n/N^{1/d}\to 0$. For continuous functions $f$ of $\mathbb {S}^{d}$ satisfying a Lipschitz condition with constant $C_{f}$ the bound is $(12dC_{f} + C_{d}’ \|f\|_{\infty }) \mathcal {O}(N^{-1/(d+2)})$ as $N\to \infty$.References
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Additional Information
- J. S. Brauchart
- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 730033
- 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)
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2398782