Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

We introduce two families of generators (functions) $\Ff$ 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, new sharp results are obtained on the density of semi-regular lattices for the Gabor frames generated by elements from these families.