Gabor frames and totally positive functions

Abstract

Let $g$ be a totally positive function of finite type, that is, $ĝ\left(\xi \right)={\prod }_{\nu =1}^M(1+2\pi i{\delta }_{\nu }\xi {)}^{-1}$ for ${\delta }_{\nu }\in \mathbb{R}$, ${\delta }_{\nu }\ne 0$, and $M\ge 2$, and let $\alpha ,\beta >0$. Then the set $\left\{e^{2\pi i\beta lt}g\right(t-\alpha k):k,l\in \mathbb{Z}\}$ is a frame for $L^2\left(\mathbb{R}\right)$ if and only if $\alpha \beta <1$. 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.