Gabor frames and totally positive functions

Let g be a totally positive function of finite type, that is, \hat{g}(\xi)=\prod_{\nu=1}^{M}(1+2\pi i\delta_{\nu}\xi)^{-1} for \delta_{\nu}\in\mathbb {R} , \delta_{\nu}\neq0 , and M\geq2 , and let \alpha,\beta\gt 0 . Then the set \{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\lt 1 . This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, 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 yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.