Stable STFT Phase Retrieval and Poincaré Inequalities

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, that is, reconstructing $ f\in L<^>{2}(\mathbb{R})$ from its spectrogram, $|\mathcal{V}_{g} f|$ where $$ \begin{align*} & \mathcal{V}_g f(x,\xi) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e<^>{-2\pi i \xi t}\,\mbox{d}t, \end{align*} $$ 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 $g(t)=e<^>{-t}{1\kern-3.4pt1}_{[0,\infty )}(t)$ and $g(t)=\exp (t-e<^>{t})$. As a by-product we establish a modified version of Poincar & eacute;'s inequality, which can be applied to non-differentiable functions and may be of independent interest.