Abstract: We study nonuniform sampling in shift-invariant spaces whose generator is a totally positive function. For a subclass of such generators the sampling theorems can be formulated in analogy to the theorems of Beurling and Landau for bandlimited functions. These results are optimal and validate the heuristic reasonings in the engineering literature. In contrast to the cardinal series, the reconstruction procedures for sampling in a shift-invariant space with a totally positive generator are local and thus accessible to numerical linear algebra.
A subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames leads to new and optimal results for Gabor frames. We show that the set of phase-space shifts of $g$ (totally positive with a Gaussian part) with respect to a rectangular lattice forms a frame, if and only if the density of the lattice is strictly larger than 1. This solves an open problem going backto Daubechies in 1990 for the class of totally positive functions of Gaussian type.