Stable STFT phase retrieval and Poincar\'e 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, i.e. reconstructing f∈ L²(R) from its spectrogram |Vg f| where Vg f(x,ξ) = ∫R f(t)g(t-x)̄e-2π i ξ t\,dt, 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-π t² 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⁻ᵗ1[0,∞)(t) and g(t)=exp(t-eᵗ). 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.