A Koksma–Hlawka inequality for general discrepancy systems

Abstract

a b s t r a c tMotivated 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.

On functions of bounded variation

“…In the following, we recall the notion of variation introduced in [20]. Let D denote an arbitrary family of measurable subsets of…”

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The recently introduced concept of D-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from [20] whether every function of bounded Hardy-Krause variation is Borel measurable and has bounded D-variation. Moreover, we show that the space of functions of bounded D-variation can be turned into a commutative Banach algebra.

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Integral Equations, Quasi-Monte Carlo Methods and Risk Modeling

“…Pausinger & Svane [25] have shown that twice continuously differentiable functions f admit finite V K (f ), and in addition they gave a bound which will be usefull in our context.…”

“…In the remaining part of the introduction we briefly sketch a more general concept of multidimensional variation which was recently developed in [25]. Let D denote an arbitrary family of measurable subsets of [0, 1] s which contains the empty set ∅ and [0, 1] s .…”

“…and A is defined to be the collection of algebraic sums of sets in D. As in [25] we define the Harman complexity h(A) of a non-empty set A ∈ A, A = [0, 1] s as the minimal number m such there exist A 1 , . .…”