On the Koksma-Hlawka inequality
On the Koksma-Hlawka inequality
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关于 Koksma-Hlawka 不等式
DOI：
10.1016/j.jco.2012.10.003
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
G. Travaglini
中科院分区：
文献类型：
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作者：
L. Brandolini;L. Colzani;G. Gigante;G. Travaglini
经典的Koksma-Hlawka不等式不适用于具有简单不连续的函数。这里我们陈述了一个kokma - hlawka型不等式，它适用于分段光滑函数fχΩ，其中f光滑和Ω是Borel子集[0,1]d：其中[公式：见文本]是所有d维区间上的最高值的差异，V(f)是总变差。我们陈述了由land lq范数测量的变化和差异的类似结果，1 /p+1/q=1，我们也给出了紧流形的扩展。
The classical Koksma–Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma–Hlawka type inequality which applies to piecewise smooth functions fχΩ, with f smooth and Ω a Borel subset of [0,1]d: where [Formula: see text] is the discrepancy the supremum is over all d-dimensional intervals, and V(f) is the total variation We state similar results with variation and discrepancy measured by Lpand Lqnorms, 1/p+1/q=1, and we also give extensions to compact manifolds.