A Koksma-Hlawka inequality for general discrepancy systems

F. Pausinger, A. Svane, Journal of Complexity 31 (2015) 773–797.

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Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2015-12-01
Journal Title
Journal of Complexity
Acknowledgement
F.P. is supported by the Graduate School of IST Austria, A.M.S is supported by the Centre for Stochastic Geometry and Advanced Bioimaging funded by a grant from the Villum Foundation.
Volume
31
Issue
6
Page
773 - 797
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Pausinger F, Svane A. A Koksma-Hlawka inequality for general discrepancy systems. Journal of Complexity. 2015;31(6):773-797. doi:10.1016/j.jco.2015.06.002
Pausinger, F., & Svane, A. (2015). A Koksma-Hlawka inequality for general discrepancy systems. Journal of Complexity, 31(6), 773–797. https://doi.org/10.1016/j.jco.2015.06.002
Pausinger, Florian, and Anne Svane. “A Koksma-Hlawka Inequality for General Discrepancy Systems.” Journal of Complexity 31, no. 6 (2015): 773–97. https://doi.org/10.1016/j.jco.2015.06.002.
F. Pausinger and A. Svane, “A Koksma-Hlawka inequality for general discrepancy systems,” Journal of Complexity, vol. 31, no. 6, pp. 773–797, 2015.
Pausinger F, Svane A. 2015. A Koksma-Hlawka inequality for general discrepancy systems. Journal of Complexity. 31(6), 773–797.
Pausinger, Florian, and Anne Svane. “A Koksma-Hlawka Inequality for General Discrepancy Systems.” Journal of Complexity, vol. 31, no. 6, Academic Press, 2015, pp. 773–97, doi:10.1016/j.jco.2015.06.002.
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