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QMC rules of arbitrary high order: Reproducing kernel Hilbert space approach. (English) Zbl 1186.65005
The aim of the paper is to show how some results of the second author can be achieved in a reproducing kernel Hilbert space setting. In particular, the authors consider numerical integration in a weighted Sobolev spaces and prove results about the worst-case error. Also they provide exact formulae and bounds for the integration errors, and present some numerical results for the test problems.

MSC:
65C05 Monte Carlo methods
11K38 Irregularities of distribution, discrepancy
11K45 Pseudo-random numbers; Monte Carlo methods
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
65D32 Numerical quadrature and cubature formulas
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