实直线和单位圆上的加权Fekete点
Weighted Fekete points on the real line and the unit circle
摘要：
加权Fekete点被定义为在固定集合上最大化范德蒙德行列式的加权版本的那些点。它们也可以看作是单位离散电荷在外静电场中的平衡分布。虽然这些点有许多应用，但它们很难明确找到，并且只在少数（未加权）经典情况下已知。我们给出了两个罕见的显式加权Fekete点的例子。第一个是为权重$w(x)=|x-ai|^{-s}$的实线，$s\ge 1$和$a\neq 0,$，而第二个是为权重$w(z)=1/|z-b|$的单位圆，与$b\in\mathbb{R},\ b\neq\pm 1.$在这两种情况下，我们提供的解决方案的连续能源问题与外部领域表示的极限版本考虑加权Fekete点的问题。
英文摘要：
Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to find explicitly, and are only known in a few (unweighted) classical cases. We give two rare explicit examples of weighted Fekete points. The first one is for the weights $w(x)=|x-ai|^{-s}$ on the real line, with $s\ge 1$ and $a\neq 0,$ while the second is for the weights $w(z)=1/|z-b|$ on the unit circle, with $b\in\mathbb{R},\ b\neq\pm 1.$ In both cases, we provide solutions of the continuous energy problems with external fields that express the limit versions of considered weighted Fekete points problems.
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