Title: Lattice rules for nonperiodic smooth integrands
Authors: Dick, Josef * #
Nuyens, Dirk * # ×
Pillichshammer, Friedrich * #
Issue Date: 2014
Publisher: Springer
Series Title: Numerische Mathematik vol:126 issue:2 pages:259-291
Abstract: 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.
ISSN: 0029-599X
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Numerical Analysis and Applied Mathematics Section
* (joint) first author
× corresponding author
# (joint) last author

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