  In this article, we consider the problem of approximating a finite set of
data (usually huge in applications) by invariant subspaces generated through a
small set of smooth functions. The invariance is either by translations under a
full-rank lattice or through the action of crystallographic groups. Smoothness
is ensured by stipulating that the generators belong to a Paley-Wiener space,
that is selected in an optimal way based on the characteristics of the given
data. To complete our investigation, we analyze the fundamental role played by
the lattice in the process of approximation.
