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We study nonuniform sampling in shift-invariant spaces and the constructionof Gabor frames with respect to the class of totally positive functions whoseFourier transform factors as $ \hat g(\xi)= \prod_{j=1}^n (1+2\pii\delta_j\xi)^{-1} \, e^{-c \xi^2}$ for $\delta_1,\ldots,\delta_n\in\mathbb{R}, c >0$ (in which case $g$ is called totally positive of Gaussiantype).  In analogy to Beurling's sampling theorem for the Paley-Wiener space ofentire functions, we prove that every separated set with lower Beurling density$>1$ is a sampling set for the shift-invariant space generated by such a $g$.In view of the known necessary density conditions, this result is optimal andvalidates the heuristic reasonings in the engineering literature.  Using a subtle connection between sampling in shift-invariant spaces and thetheory of Gabor frames, we show that the set of phase-space shifts of $g$ withrespect to a rectangular lattice $\alpha \mathbb{Z} \times \beta \mathbb{Z}$forms a frame, if and only if $\alpha \beta <1$. This solves an open problemgoing back to Daubechies in 1990 for the class of totally positive functions ofGaussian type.  The proof strategy involves the connection between sampling inshift-invariant spaces and Gabor frames, a new characterization of samplingsets "without inequalities" in the style of Beurling, new properties of totallypositive functions, and the interplay between zero sets of functions in ashift-invariant space and functions in the Bargmann-Fock space.

Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions