Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces

We extend the notion of L2L2–BB-discrepancy introduced in [E. Novak, H. Woźniakowski, L2L2 discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359–388] to what we shall call weighted geometric L2L2-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L2L2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Woźniakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.