论文标题
单位圆和球上贪婪的能量序列的渐近级
Asymptotics of greedy energy sequences on the unit circle and the sphere
论文作者
论文摘要
对于一个参数$λ> 0 $，我们研究了单位球$ s^{d} \ s^{d} \ subset \ subset \ subbb {r}^r}^r}^r}^r}^r}^^d+1} $，$ s $ s $ n = 0}^{\ infty} $ _ {a_ {n}）_ {a_ {n}） $ a_ {n} $，$ n \ geq 1 $，是一个潜在的$ \ sum_ {k = 0}^{n-1} | x-a_ {k} |^λ$达到$ s^{d} $的最大值。我们表明，这些序列满足对称属性$ a_ {2k+1} = - a_ {2k} $对于每个$ k \ geq 0 $。序列的渐近分布在值$λ= 2 $的情况下进行急剧过渡，从均匀分布（$λ<2 $）到两个反物点（$λ> 2 $）的集中度。我们研究了序列第一$ n $点的$λ$ - 能量的一阶和二阶渐近学，以及极值$ \ sum_ {k = 0}^{n-1}^{n-1} | a_ {n} -a__ {k} -a_ {k} |^λ$的渐近行为。在单位圆上分析了二阶渐近学。结果表明，这种渐近行为与单位圆上的$ n $均等点明显不同，并且行为的过渡发生在$λ= 1 $。
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 $λ=1$.
