Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces
 Title:

Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces
 Author(s):

Gnewuch, Michael
 Date:

2010
 Type:

Technical reports
 Department:

Computer Science
 Persistent URL:

http://hdl.handle.net/10022/AC:P:10523
 Series:

Columbia University Computer Science Technical Reports
 Part Number:

CUCS03410
 Publisher:

Department of Computer Science, Columbia University
 Publisher Location:

New York
 Abstract:

We extend the notion of L2Bdiscrepancy introduced in [E. Novak, H. Wozniakowski, L2 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 what we want to call weighted geometric L2discrepancy. This extended notion allows us to consider weights to moderate the importance of different groups of variables, and additionally volume measures different from the Lebesgue measure as well as classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L2discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski. 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 algorithms.
 Subject(s):

Computer science
 Item views
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 Metadata:

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 Suggested Citation:

Michael Gnewuch,
2010,
Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces, Columbia University Academic Commons,
http://hdl.handle.net/10022/AC:P:10523.