QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach |
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Authors: | Jan Baldeaux Josef Dick |
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Affiliation: | 1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
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Abstract: | In this paper we consider numerical integration of smooth functions lying in a particular reproducing kernel Hilbert space. We show that the worst-case error of numerical integration in this space converges at the optimal rate, up to some power of a log?N factor. A similar result is shown for the mean square worst-case error, where the bound for the latter is always better than the bound for the square worst-case error. Finally, bounds for integration errors of functions lying in the reproducing kernel Hilbert space are given. The paper concludes by illustrating the theory with numerical results. |
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