Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters

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.