离散测度的有限点集逼近
Approximation of Discrete Measures by Finite Point Sets
摘要：
对于概率测度$μ$在$[0,1]$上没有离散分量，由一个有限点集合根据星差的最佳可能逼近阶是$\frac{1}{2N}$，这是最近已经证明的.然而，如果$μ$包含离散分量，则一般不存在非平凡下限，因为在这种情况下，构造示例是简单的，没有任何近似误差。这也许可以解释，为什么$[0,1]$上的离散措施的逼近有限点集迄今尚未完全涵盖在现有的文献。在这篇文章中，我们通过对离散情况的完整描述来弥补这一差距。最重要的是，我们证明了，对于任何离散措施的最佳可能的近似顺序是无穷多$N$有界从下面的$\frac{1}{cN}$的一些常数$c \geq 2$，这取决于措施。这意味着，对于$[0,1]^d$上的一个支持的离散测度，已知的可能的近似阶数$\frac{1}{N}$确实是最优的。
英文摘要：
For a probability measure $μ$ on $[0,1]$ without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is $\frac{1}{2N}$ as has been proven relatively recently. However, if $μ$ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on $[0,1]$ by finite point sets has so far not been completely covered in the existing literature. In this note, we close this gap by giving a complete description of the discrete case. Most importantly, we prove that for any discrete measure the best possible order of approximation is for infinitely many $N$ bounded from below by $\frac{1}{cN}$ for some constant $c \geq 2$ which depends on the measure. This implies, that for a finitely supported discrete measure on $[0,1]^d$ the known possible order of approximation $\frac{1}{N}$ is indeed the optimal one.
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