贪婪能点的渐近性
Asymptotics of Greedy Energy Points
摘要：
对于局部紧Hausdorff空间$X$上的对称核$k:X\times X \to \mathbb{R}\cup\{+\infty\}$，我们研究了通过任意选择$a_{1}\in A$和$a_{n+1}$使得$\sum_{i=1}^{n}k(a_{n+1},a_{i})=\inf_{x\in A}\sum_{i=1}^{n}k(x,a_{i})$来归纳定义的紧子集$A\subset X$的贪婪$k$ -能量点$\{a_{i}\}_{1}^{\infty}$的渐近行为.我们给出了这些点（也称为Leja点）渐近能量最小化的充分条件（即具有能量$\sum_{i\neq j}^{N}k(a_{i},a_{j})$作为$N\to\infty$，与$\mathcal{E}(A,N):=\min\{\sum_{i\neq j}k(x_{i},x_{j}):x_{1},...,x_{N}\in A\}$渐近相同），并且具有与$A$的平衡测度相等的渐近分布。对于Riesz内核的情况下，$k_{s}(x,y):=|x-y|^{-s}$，$s>0$，我们表明，如果$A$是一个可求长的约旦弧或封闭曲线在$\mathbb{R}^{p}$和$s>1$，那么贪婪的$k_{s}$能量点不渐近能量最小化，在相反的情况下$s<1$。(In事实上，我们表明，没有一个点序列可以渐近能量最小化$s>1$。贪婪$k_{s}$能量点上的一个领域，贪婪的最佳包装点，加权Riesz内核获得额外的结果。
英文摘要：
For a symmetric kernel $k:X\times X \to \mathbb{R}\cup\{+\infty\}$ on a locally compact Hausdorff space $X$, we investigate the asymptotic behavior of greedy $k$-energy points $\{a_{i}\}_{1}^{\infty}$ for a compact subset $A\subset X$ that are defined inductively by selecting $a_{1}\in A$ arbitrarily and $a_{n+1}$ so that $\sum_{i=1}^{n}k(a_{n+1},a_{i})=\inf_{x\in A}\sum_{i=1}^{n}k(x,a_{i})$. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy $\sum_{i\neq j}^{N}k(a_{i},a_{j})$ as $N\to\infty$ that is asymptotically the same as $\mathcal{E}(A,N):=\min\{\sum_{i\neq j}k(x_{i},x_{j}):x_{1},...,x_{N}\in A\}$), and have asymptotic distribution equal to the equilibrium measure for $A$. For the case of Riesz kernels $k_{s}(x,y):=|x-y|^{-s}$, $s>0$, we show that if $A$ is a rectifiable Jordan arc or closed curve in $\mathbb{R}^{p}$ and $s>1$, then greedy $k_{s}$-energy points are not asymptotically energy minimizing, in contrast to the case $s<1$. (In fact we show that no sequence of points can be asymptotically energy minimizing for $s>1$.) Additional results are obtained for greedy $k_{s}$-energy points on a sphere, for greedy best-packing points, and for weighted Riesz kernels.
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