Numerische Mathematik

, Volume 126, Issue 2, pp 259–291 | Cite as

Lattice rules for nonperiodic smooth integrands

  • Josef Dick
  • Dirk Nuyens
  • Friedrich Pillichshammer


The aim of this paper is to show that one can achieve convergence rates of \(N^{-\alpha + \delta }\) for \(\alpha > 1/2\) (and for \(\delta > 0\) arbitrarily small) for nonperiodic \(\alpha \)-smooth cosine series using lattice rules without random shifting. The smoothness of the functions can be measured by the decay rate of the cosine coefficients. For a specific choice of the parameters the cosine series space coincides with the unanchored Sobolev space of smoothness 1. We study the embeddings of various reproducing kernel Hilbert spaces and numerical integration in the cosine series function space and show that by applying the so-called tent transformation to a lattice rule one can achieve the (almost) optimal rate of convergence of the integration error. The same holds true for symmetrized lattice rules for the tensor product of the direct sum of the Korobov space and cosine series space, but with a stronger dependence on the dimension in this case.

Mathematics Subject Classification

65D30 65D32 11K16 11K36 



J.D. is supported by an Australian Research Council Queen Elizabeth II fellowship. D.N. is a fellow of the Research Foundation Flanders (FWO) and thanks Prof. Ian H. Sloan for initial discussions on the half-period cosine space. The first two authors are grateful to the Hausdorff Institute in Bonn where most of this research was carried out. F.P. is partially supported by the Austrian Science Foundation (FWF), Project S9609.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Josef Dick
    • 1
  • Dirk Nuyens
    • 2
  • Friedrich Pillichshammer
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Computer ScienceKU LeuvenHeverleeBelgium
  3. 3.Institut für FinanzmathematikUniversität LinzLinzAustria

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