Asymptotics of greedy energy sequences on the unit circle and the sphere

Abstract

For a parameter $\lambda >0$, we investigate greedy $\lambda$-energy sequences $(a_n{)}_{n=0}^{\infty }$ on the unit sphere $S^d\subset {\mathbb{R}}^{d+1}$, $d\ge 1$, satisfying the defining property that each $a_n$, $n\ge 1$, is a point where the potential ${\sum }_{k=0}^{n-1}|x-a_k{|}^{\lambda }$ attains its maximum value on $S^d$. We show that these sequences satisfy the symmetry property $a_{2k+1}=-a_{2k}$ for every $k\ge 0$. The asymptotic distribution of the sequence undergoes a sharp transition at the value $\lambda =2$, from uniform distribution ($\lambda <2$) to concentration on two antipodal points ($\lambda >2$). We investigate first-order and second-order asymptotics of the $\lambda$-energy of the first $N$ points of the sequence, as well as the asymptotic behavior of the extremal values ${\sum }_{k=0}^{n-1}|a_n-a_k{|}^{\lambda }$. The second-order asymptotics is analyzed on the unit circle. It is shown that this asymptotic behavior differs significantly from that of $N$ equally spaced points on the unit circle, and a transition in the behavior takes place at $\lambda =1$.

35 pages, 6 figures, this is the published version

Cited by in corpus (1)

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