Polarization

B. Elliott
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引用次数: 0

Abstract

. We investigate the behavior of a greedy sequence on the sphere S d defined so that at each step the point that minimizes the Riesz s -energy is added to the existing set of points. We show that for 0 < s < d , the greedy sequence achieves optimal second-order behavior for the Riesz s -energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz s -kernels is of order N s/d in the same range 0 < s < d . Furthermore, using the Stolarsky principle relating the L 2 -discrepancy of a point set with the pairwise sum of distances (Riesz energy with s = − 1), we also obtain a simple upper bound on the L 2 -spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.
极化
. 我们研究了一个贪心序列在球面上的行为,该贪心序列定义为在每一步中将Riesz S -能量最小的点添加到现有的点集中。我们证明了当0 < s < d时,贪心序列对于Riesz s -能量(不超过常数)达到最优的二阶行为。为了得到这个结果,我们证明了具有Riesz s -核的最大偏振的二阶项在0 < s < d范围内为N s/d阶。此外,利用将点集的l2 -差异与距离的两两和(s = - 1的Riesz能量)联系起来的Stolarsky原理,我们也得到了贪心序列的l2 -球帽差异的一个简单上界,并给出了数值例子,表明真正的差异要小得多。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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