Asymptotics of Greedy Energy Points
A. L\'opez Garc\'ia, E. B. Saff

TL;DR
This paper studies the asymptotic behavior of greedy energy points, known as Leja points, on compact sets, providing conditions for their energy optimality and analyzing specific cases like Riesz kernels.
Contribution
It establishes sufficient conditions for greedy energy points to be asymptotically energy minimizing and describes their distribution, with new insights for Riesz kernels on various geometries.
Findings
Greedy points are asymptotically energy minimizing under certain conditions.
For Riesz kernels with s>1 on curves, greedy points are not energy minimizing.
On spheres and for weighted kernels, additional asymptotic behaviors are characterized.
Abstract
For a symmetric kernel on a locally compact Hausdorff space , we investigate the asymptotic behavior of greedy -energy points for a compact subset that are defined inductively by selecting arbitrarily and so that . We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy as that is asymptotically the same as ), and have asymptotic distribution equal to the equilibrium measure for . For the case of Riesz kernels , , we show that if is a rectifiable Jordan arc or closed…
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Taxonomy
TopicsMathematical Approximation and Integration · Mathematical functions and polynomials · Markov Chains and Monte Carlo Methods
