  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.
