关于举行戴欣荣教授（中山大学）学术报告会的通知

报告主题:    Frame set for Gabor systems with Haar window

报 告人: 戴欣 荣教授（中山大学）

报告时间：2022年 10月7日（星期五）下午3:30-4:30

报告地点：四号楼224室

邀 请人: 李兵 教授

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数学学院

2022年10月4日

报告摘要：We show the full structure of the frame set for the  Gabor system $\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. This is the first compactly supported window function that the frame set is represented explicitly. The strategy of the proof  is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to  provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish  {a} necessary and sufficient condition for the Gabor system $\mathcal{G}(g;\alpha,\beta)$ to be a frame, i.e.,  the symmetric invariant set of the transformation $\mathcal{M}$ is empty.