Totally Positive Functions and Gabor Frames over Rational Lattices

We show that for an arbitrary totally positive function $g\in L^1(\mathbb{R})$ and $\alpha \beta$ rational, the Gabor family $\{e^{2\pi i\beta lt}g(t-\alpha k):k,l\in \mathbb{Z}\}$ is a frame for $L^2(\mathbb{R})$, if and only if $\alpha \beta <1$.