Stable STFT phase retrieval and Poincaré inequalities
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929404265889792 |
|---|---|
| author | Rathmair, Martin |
| author_facet | Rathmair, Martin |
| contents | 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 $$\mathcal{V}_g f(x,ξ) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2πi ξt}\,\mbox{d}t,$$ have been classified in terms of the connectivity of the measurements. These findings were however crucially restricted to the case where the window $g(t)=e^{-πt^2}$ 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 Poincaré's inequality which can be applied to non-differentiable functions and may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable STFT phase retrieval and Poincaré inequalities Rathmair, Martin Functional Analysis Classical Analysis and ODEs 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 $$\mathcal{V}_g f(x,ξ) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2πi ξt}\,\mbox{d}t,$$ have been classified in terms of the connectivity of the measurements. These findings were however crucially restricted to the case where the window $g(t)=e^{-πt^2}$ 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 Poincaré's inequality which can be applied to non-differentiable functions and may be of independent interest. |
| title | Stable STFT phase retrieval and Poincaré inequalities |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.00398 |