Title: Stable STFT phase retrieval and Poincaré inequalities Show Chinese title

Title: 稳定STFT相位恢复与Poincaré不等式

Abstract: In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] and [P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate functions. Journal of the European Mathematical Society (2021)] the instabilities of Gabor phase retrieval problem, i.e. reconstructing $ f\in L^2(\mathbb{R})$ from its spectrogram $|\mathcal{V}_g f|$ where $$\mathcal{V}_g f(x,\xi) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2\pi i \xi t}\,\mbox{d}t,$$ have been classified in terms of the connectivity of the measurements. These findings were however crucially restricted to the case where the window $g(t)=e^{-\pi t^2}$ is Gaussian. In this work we establish a corresponding result for a number of other window functions including the one-sided exponential $g(t)=e^{-t}\mathbb{1}_{[0,\infty)}(t)$ and $g(t)=\exp(t-e^t)$. As a by-product we establish a modified version of Poincar\'e's inequality which can be applied to non-differentiable functions and may be of independent interest. Show Chinese abstract

Abstract: 在最近的工作 [P. Grohs 和 M. Rathmair. 稳定的 Gabor 相位检索与谱聚类.《纯数学和应用数学通讯》(2018)] 和 [P. Grohs 和 M. Rathmair. 多变量函数的稳定 Gabor 相位检索.《欧洲数学会杂志》(2021)] 中，研究了 Gabor 相位检索问题的不稳定性，即从信号的光谱图$|\mathcal{V}_g f|$中重构$ f\in L^2(\mathbb{R})$，其中窗口函数$$\mathcal{V}_g f(x,\xi) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2\pi i \xi t}\,\mbox{d}t,$$的不稳定性已根据测量值的连通性进行了分类。然而，这些发现仅限于窗口函数$g(t)=e^{-\pi t^2}$为高斯窗的情况。在本文中，我们针对若干其他窗口函数（包括单侧指数窗$g(t)=e^{-t}\mathbb{1}_{[0,\infty)}(t)$和$g(t)=\exp(t-e^t)$）建立了相应的结果。作为副产品，我们还建立了一个修改版的庞加莱不等式，它可以应用于不可微函数，并且可能具有独立的研究兴趣。