Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

Abstract

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_n{\}}_{n=1}^{\infty }$ on the unit sphere ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1,x_2,...,x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2\le s<d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

26 pages