  This paper provides the theoretical foundation for the construction of
lattice algorithms for multivariate $L_2$ approximation in the worst case
setting, for functions in a periodic space with general weight parameters. Our
construction leads to an error bound that achieves the best possible rate of
convergence for lattice algorithms. This work is motivated by PDE applications
in which bounds on the norm of the functions to be approximated require special
forms of weight parameters (so-called POD weights or SPOD weights), as opposed
to the simple product weights covered by the existing literature. Our result
can be applied to other lattice-based approximation algorithms, including
kernel methods or splines.
