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Asymptotics of greedy energy points

Authors: A. López García and E. B. Saff
Journal: Math. Comp. 79 (2010), 2287-2316
MSC (2010): Primary 65D99, 52A40; Secondary 78A30
Published electronically: April 16, 2010
MathSciNet review: 2684365
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Abstract: For a symmetric kernel $ k:X\times X \rightarrow \mathbb{R}\cup\{+\infty\}$ on a locally compact metric space $ X$, we investigate the asymptotic behavior of greedy $ k$-energy points $ \{a_{i}\}_{1}^{\infty}$ for a compact subset $ A\subset X$ that are defined inductively by selecting $ a_{1}\in A$ arbitrarily and $ a_{n+1}$ so that $ \sum_{i=1}^{n}k(a_{n+1},a_{i})=\inf_{x\in A}\sum_{i=1}^{n}k(x,a_{i})$. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy $ \sum_{i\neq j}^{N}k(a_{i},a_{j})$ as $ N\rightarrow\infty$ that is asymptotically the same as $ \mathcal{E}(A,N):=\min\{\sum_{i\neq j}k(x_{i},x_{j}):x_{1},\ldots,x_{N}\in A\}$), and have asymptotic distribution equal to the equilibrium measure for $ A$. For the case of Riesz kernels $ k_{s}(x,y):=\vert x-y\vert^{-s}$, $ s>0$, we show that if $ A$ is a rectifiable Jordan arc or closed curve in $ \mathbb{R}^{p}$ and $ s>1$, then greedy $ k_{s}$-energy points are not asymptotically energy minimizing, in contrast to the case $ s<1$. (In fact, we show that no sequence of points can be asymptotically energy minimizing for $ s>1$.) Additional results are obtained for greedy $ k_{s}$-energy points on a sphere, for greedy best-packing points (the case $ s=\infty$), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.

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Additional Information

A. López García
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

E. B. Saff
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Minimal energy, Leja points, equilibrium measure, Riesz kernels, best-packing configurations, Voronoi cells
Received by editor(s): December 12, 2008
Received by editor(s) in revised form: June 27, 2009
Published electronically: April 16, 2010
Additional Notes: The results of this paper form a part of the first author’s Ph.D. dissertation at Vanderbilt University
The research of the second author was supported, in part, by National Science Foundation grants DMS-0603828 and DMS-0808093.
Article copyright: © Copyright 2010 A. López García and E. B. Saff

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