Gabor frames for rational functions

Let $g$ be a function from $L^2(\mathbb{R})$. With every $\alpha,\beta>0$ we connect the Gabor system $G(g,\alpha,\beta)$ of time-frequency shifts of $g$,
$$ G(g,\alpha,\beta)=\{e^{2\pi i \alpha m x}g(x-\beta n)\}_{m,n\in\mathbb{Z}}. $$
The main question of Gabor analysis is to describe frame set, i.e. to describe pairs $\alpha,\beta$ such that system $G(g,\alpha,\beta)$ generates a frame in $L^2(\mathbb{R})$.
Up to now it was known only few functions $g$ with complete description of frame set. The answer has been obtained for the Gaussian (Lyubarskii, Seip), truncated and symmetric exponential functions (Jannsen), the hyperbolic secant (Jannsen). Despite numerous efforts little progress has been done until 2011. A breakthrough was achieved by Grochenig, Romero and Stockler who considered the class of totally positive functions of finite type and, by using another approach, Gaussian totally positive functions of finite type.
We managed to find a new class of functions with complete description of frame set – rational functions of Herglotz type. This was done by combination of classical theory of entire functions with some ideas from dynamical systems. We also proved some other results for arbitrary rational functions and some results for non-lattice Gabor systems.
The talk is based on joint works with A. Kulikov and Yu. Lyubarskii.