DagSemProc.09391.5.pdf
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Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces
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Dagstuhl Seminar Proceedings, Volume 9391
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
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- Publication Date: 2009-12-02
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Michael Gnewuch. Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 9391, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)
https://doi.org/10.4230/DagSemProc.09391.5
Abstract
We extend the notion of $L_2$ $B$ discrepancy provided in
[E. Novak, H. Wo'zniakowski, $L_2$ discrepancy and multivariate
integration, in: Analytic number theory. Essays in honour of Klaus
Roth. 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 Lebesgue
measure and classes of test sets different from measurable subsets
of some Euclidean space.
We relate the weighted $L_2$ $mathcal{B}$ discrepancy to numerical
integration defined over weighted reproducing kernel Hilbert spaces
and settle in this way an open problem posed by Novak and
Wo'zniakowski.
Subject Classification
Keywords
- Discrepancy
- Numerical Integration
- Quasi-Monte Carlo
- Reproducing Kernel Hilbert Space
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