A Koksma-Hlawka inequality for general discrepancy systems

作者：Pausinger Florian^*; Svane Anne Marie

来源：Journal of Complexity, 2015, 31(6): 773-797.

DOI：10.1016/j.jco.2015.06.002

• Harman variation
• Hardy-Krause variation
• Koksma-Hlawka theorem
• Integral geometry

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 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. 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-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.

• 出版日期2015-12