【学术报告】Gabor analysis for rational functions

• 主讲人：Prof. Yuri Belov
• 时间：Dec. 2, 2021 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)
• 地点：online

Abstract:   Let $g$ be a function in $L^2(\mathbb{R})$. By $G_\Lambda$, $\Lambda\subset R^2$  we denote the system of time-frequency shifts of $g$, $G_\Lambda=\{e^{2\pi i \omega x}g(x-t)\}_{(t,\omega)\in\Lambda}$.

A typical  model set $\Lambda$ is the rectangular lattice  $\Lambda_{\alpha, \beta}:= \alpha\mathbb{Z}\times\beta\mathbb{Z}$  and one of the basic  problems of the Gabor analysis is the description of the frame set of $g$ i.e., all pairs $\alpha, \beta$ such that  $G_\Lambda_{\alpha,\beta}$ is a frame   in $L^2(\mathbb{R})$. It follows from the general theory that $\alpha\beta \leq 1$ is a necessary condition (we assume $\alpha, \beta > 0$, of course). Do all such $\alpha, \beta $  belong to the frame set of $g$? Up to 2011 only few such functions $g$ (up to translation, modulation, dilation and Fourier transform) were known. In 2011  K. Grochenig and J. Stockler extended this class by including the totally positive functions of finite type(uncountable family yet depending on finite number of parameters) and later added the Gaussian finite type totally positive functions. We suggest another approach to the problem and prove that  all Herglotz rational functions with imaginary poles  also belong to this class. This  approach also gives  new results for general rational functions.  In particular, we are able to confirm Daubechies conjecture for rational functions and irrational densities.