  We introduce two families of generators (functions) $\mathcal{G}$ that
consist of entire and meromorphic functions enjoying a certain periodicity
property and contain the classical Gaussian and hyperbolic secant generators.
Sharp results are proved on the density of separated sets that provide
non-uniform sampling for the shift-invariant and quasi shift-invariant spaces
generated by elements of these families. As an application, we obtain new sharp
results on the density of semi-regular lattices for the Gabor frames generated
by elements from these families.
