Date:
Wed, 15/03/202314:00-15:00
Title: Gabor frames for rational functions
Abstract: One of the most interesting systems in time-frequency analysis are the so-called Gabor systems: we take the function $g\in L^2(\mathbb{R})$ and consider its time-frequency shifts with respect to the lattice $\alpha \mathbb{Z} \times \beta\mathbb{Z}$, $\alpha, \beta > 0$
$$G(g;\alpha, \beta) = \{e^{2\pi i m \beta x}g(x-\alpha n)\}_{n, m\in \mathbb{Z}}$$
We are interested in all pairs $(\alpha, \beta)$ for which this system is a frame, that is there are some constants $A, B>0$ such that for all $f\in L^2(\mathbb{R})$ we have
$$A||f||_{L^2(\mathbb{R})}^2 \le \sum_{n, m\in\Z} |\langle f, e^{2\pi i m \beta x}g(x-\alpha n)\rangle|^2 \le B||f||_{L^2(\mathbb{R})}^2.$$
General theory of Gabor frames says that for this to hold $\alpha\beta$ should necessarily be at most $1$, and moreover if the function $g$ decays fast enough we must have $\alpha\beta < 1$. Until recently, complete description of the pairs $(\alpha, \beta)$ was known only for a handful of functions. In 1992 Lyubarskii and Seip covered the case of the Gaussian, in 1996 and 2003 Janssen solved one-sided and two-sided exponentials respectively, and in 2002 Janssen and Strohmer solved the case of the hyperbolic secant. In all these cases the frames set turned out to be either $\{\alpha, \beta > 0: \alpha\beta \le 1\}$ or $\{\alpha, \beta > 0: \alpha\beta < 1\}$.
In 2011, a breakthrough was achieved by Gröchenig and Stöckler showing that all pairs with $\alpha\beta < 1$ give a frame for the functions $g$ which are totally positive of finite type, and in 2017 Gröchenig, Romero and Stöckler extended this to the Gaussian totally positive functions of finite type, thus for the first time giving an infinite, multi-parameter set of functions for which the full answer is known.
I will tell about the recent progress made in the case when the function $g(x) = \frac{P(x)}{Q(x)}$ is an arbitrary rational function. Among the results are complete characterization in the cases when $g$ is a Herglotz rational function and when $\deg\, Q(x) \le 2$. In addition, we prove that for almost all rational functions all pairs $(\alpha, \beta)$ with $\alpha\beta < 1$ and $\alpha\beta
otin \Q$ give us a frame.
The talk is based on a joint works with Yurii Belov and Yurii Lyubarskii.
Abstract: One of the most interesting systems in time-frequency analysis are the so-called Gabor systems: we take the function $g\in L^2(\mathbb{R})$ and consider its time-frequency shifts with respect to the lattice $\alpha \mathbb{Z} \times \beta\mathbb{Z}$, $\alpha, \beta > 0$
$$G(g;\alpha, \beta) = \{e^{2\pi i m \beta x}g(x-\alpha n)\}_{n, m\in \mathbb{Z}}$$
We are interested in all pairs $(\alpha, \beta)$ for which this system is a frame, that is there are some constants $A, B>0$ such that for all $f\in L^2(\mathbb{R})$ we have
$$A||f||_{L^2(\mathbb{R})}^2 \le \sum_{n, m\in\Z} |\langle f, e^{2\pi i m \beta x}g(x-\alpha n)\rangle|^2 \le B||f||_{L^2(\mathbb{R})}^2.$$
General theory of Gabor frames says that for this to hold $\alpha\beta$ should necessarily be at most $1$, and moreover if the function $g$ decays fast enough we must have $\alpha\beta < 1$. Until recently, complete description of the pairs $(\alpha, \beta)$ was known only for a handful of functions. In 1992 Lyubarskii and Seip covered the case of the Gaussian, in 1996 and 2003 Janssen solved one-sided and two-sided exponentials respectively, and in 2002 Janssen and Strohmer solved the case of the hyperbolic secant. In all these cases the frames set turned out to be either $\{\alpha, \beta > 0: \alpha\beta \le 1\}$ or $\{\alpha, \beta > 0: \alpha\beta < 1\}$.
In 2011, a breakthrough was achieved by Gröchenig and Stöckler showing that all pairs with $\alpha\beta < 1$ give a frame for the functions $g$ which are totally positive of finite type, and in 2017 Gröchenig, Romero and Stöckler extended this to the Gaussian totally positive functions of finite type, thus for the first time giving an infinite, multi-parameter set of functions for which the full answer is known.
I will tell about the recent progress made in the case when the function $g(x) = \frac{P(x)}{Q(x)}$ is an arbitrary rational function. Among the results are complete characterization in the cases when $g$ is a Herglotz rational function and when $\deg\, Q(x) \le 2$. In addition, we prove that for almost all rational functions all pairs $(\alpha, \beta)$ with $\alpha\beta < 1$ and $\alpha\beta
otin \Q$ give us a frame.
The talk is based on a joint works with Yurii Belov and Yurii Lyubarskii.