Ryan E. McCleary | Searcharxiv

arXiv2021

Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

In this work we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if $(a_n)_{n=0}^{\infty}$ is a greedy $s$-energy sequence on the unit circle, the Riesz potential $U_{N,s}(x):=\sum_{k=0}^{N-1}|a_k-x|^{-s}$, $s>0$, generated by the first $N$ points of the sequence attains its minimum value at the point $a_{N}$, for every $N\geq 1$. In the case $s=0$ we minimize instead the logarithmic potential $U_{N,0}(x):=-\sum_{k=0}^{N-1}\log |a_{k}-x|$. We analyze the asymptotic properties of these extremal values $U_{N,s}(a_N)$, studying separately the cases $s=0$, $0 1$. We obtain second-order asymptotic formulas for $U_{N,s}(a_N)$ in the cases $s=0$, $0 1$ is proved, and it is shown that the normalized sequence $U_{N,s}(a_N)/N^s$ is bounded and divergent in this case. We also consider, briefly, greedy energy sequences in which the minimization condition is required starting from the point $a_{p+1}$ (instead of the point $a_{1}$ as previously stated), for some $p\geq 1$. For this more general class of greedy sequences, we prove a first-order asymptotic result for $0\leq s<1$.

math.CA↗

arXiv2021

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

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

math.CA↗