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URN: urn:nbn:de:0030-drops-22966
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2296/
Gnewuch, Michael

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

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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.

BibTeX - Entry

@InProceedings{gnewuch:DSP:2009:2296,
  author =	{Michael Gnewuch},
  title =	{Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  year =	{2009},
  editor =	{Thomas M{\"u}ller-Gronbach and Leszek Plaskota and Joseph. F. Traub},
  number =	{09391},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2296},
  annote =	{Keywords: Discrepancy, Numerical Integration, Quasi-Monte Carlo, Reproducing Kernel Hilbert Space}
}

Keywords: Discrepancy, Numerical Integration, Quasi-Monte Carlo, Reproducing Kernel Hilbert Space
Seminar: 09391 - Algorithms and Complexity for Continuous Problems
Issue date: 2009
Date of publication: 02.12.2009

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