Sharp sampling theorems in shift-invariant spaces

We present our recent work on sampling on shift-invariant spaces generated by a totally positive function of Gaussian type, discuss the main techniques, provide context for the results, and derive some new consequences. The main result is a sharp density condition for sampling that parallels Beurling's theory for bandlimited functions. As an application, we completely characterize the Gabor frame set for totally positive windows of Gaussian type.