论文标题
相似措施的家庭
Families of well approximable measures
论文作者
论文摘要
我们提供了一种算法，以通过度量$ν_{n} $对应于$ n $点的一组$ν_{n} $近似有限支持的离散度量$μ$，以使$μ$和$ν_n$之间的总变化具有上限。结果，如果$μ$是$ [0,1]^{d} $对$ [0,1]^{d} $的离散概率度量的，则每个点的重量的衰减速率足够，则$μ$可以由$ν_n$近似于总变化，并在$ nivescrepancy上，从而在$（$ log log n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n）在离散的情况下，我们的结果改善了Aistleitner，Bilyk和Nikolov的最新工作，他们表明，对于任何标准化的Borel量度$μ$，存在有限套件，其相对于$μ$的星票最多是$ $（\ log log n）此外，在情况下，我们缩小了文献差异的差异，以$ d = 1 $，表明勒贝格确实是有限集的最难衡量标准，而且所有没有离散组件的措施都与勒布斯格量度相同的差异顺序。
We provide an algorithm to approximate a finitely supported discrete measure $μ$ by a measure $ν_{N}$ corresponding to a set of $N$ points so that the total variation between $μ$ and $ν_N$ has an upper bound. As a consequence if $μ$ is a (finite or infinitely supported) discrete probability measure on $[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then $μ$ can be approximated by $ν_N$ with total variation, and hence star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure $μ$, there exist finite sets whose star-discrepancy with respect to $μ$ is at most $(\log N)^{d-\frac{1}{2}} N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case $d=1$ showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.
