A Koksma-Hlawka inequality for general discrepancy systems

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• 作者：Florian Pausinger^a ; ^{florian.pausinger@ist.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Anne Marie Svane^b ; ^{amsvane@math.au.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Harman variation ; Hardy&ndash ; Krause variation ; Koksma&ndash ; Hlawka theorem ; Integral geometry
• 刊名：Journal of Complexity
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：31
• 期：6
• 页码：773-797
• 全文大小：478 K

文摘

Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D$\mathcal{D}$ of subsets. We prove a general version of the Koksma–Hlawka theorem that holds for this notion of variation and discrepancy with respect to D$\mathcal{D}$. As special cases, we obtain Koksma–Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma–Hlawka theorem. We show that the space of functions of bounded D$\mathcal{D}$-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.