Stable STFT phase retrieval and Poincaré inequalities

2024/06/29 by Rathmair, Martin · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2407.00398

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∈ L²(ℝ) from its spectrogram |V_g f| where V_g f(x,ξ) = ∫_ℝ f(t)g(t-x)e^{-2πi ξt} 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^-πt2 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\mathbb1_[0,∞)(t) and g(t)=exp(t-e^t). As a by-product we establish a modified version of Poincaré's inequality which can be applied to non-differentiable functions and may be of independent interest.

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