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

Author Michael Gnewuch

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Keywords
  • Discrepancy
  • Numerical Integration
  • Quasi-Monte Carlo
  • Reproducing Kernel Hilbert Space

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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.

Cite As Get BibTex

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

Author Details

Michael Gnewuch
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