Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces

摘要

We extend the notion of $L_2$ $B$ discrepancy provided in [E. Novak, H. Wou27zniakowski, $L_2$ discrepancy and multivariateintegration, in: Analytic number theory. Essays in honour of KlausRoth. W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt,and R. C. Vaughan (Eds.), Cambridge University Press, Cambridge,2009, 359 -- 388] to the weighted $L_2$ $mathcal{B}$ discrepancy.This newly defined notion allows to consider weights, but also volume measures different from the Lebesguemeasure and classes of test sets different from measurable subsetsof some Euclidean space.We relate the weighted $L_2$ $mathcal{B}$ discrepancy to numericalintegration defined over weighted reproducing kernel Hilbert spacesand settle in this way an open problem posed by Novak and Wou27zniakowski.