Gabor Frames and Totally Positive Functions

Let $g$ be a totally positive function of finite type. Then the Gabor set $\{e^{2\pi i\beta lt}g(t-\alpha k),k,l\in Z\}$ is a frame for $L^2(R)$, if and only if $\alpha \beta <1$. This result is a first positive contribution to a conjecture of I.\ Daubechies from 1990. So far the complete characterization of lattice parameters $\alpha ,\beta$ that generate a frame has been known for only six window functions $g$. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.