论文标题
通过有限点集的离散度量近似
Approximation of Discrete Measures by Finite Point Sets
论文作者
论文摘要
对于$ [0,1] $上的概率度量$μ$，而无需离散组件，按照恒星票证为$ \ frac {1} {2n} $，由有限点设置的最佳近似顺序如最近相对较新。但是，如果$μ$包含一个离散的组件，则一般没有非平凡的下限持有，因为在这种情况下，它可以简单地构造示例而没有任何近似误差。这可能解释了为什么有限点集对$ [0,1] $的离散措施的近似迄今尚未在现有文献中完全涵盖。在本说明中，我们通过对离散案例进行完整描述来缩小这一差距。最重要的是，我们证明，对于任何离散的度量，最佳近似顺序是针对从下面界定的许多$ n $，$ \ frac {1} {cn} {cn} {cn} $对于某些常数$ c \ geq 2 $，这取决于该度量。这意味着，对于$ [0,1]^d $有限支持的离散度量，已知的近似值顺序$ \ frac {1} {n} $确实是最佳的。
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.
