PreprintCC BY 4.0arXiv
Phase retrieval from a uniformly discrete point set
Abstract
We prove that for every window w of the form w(x) = e−π|x|2 h(x), where h is a polynomial, there exists a uniformly discrete set of points 𝒮 ⊂ ℝ2d such that the magnitude of the short-time Fourier transform Vw f on 𝒮 determines every f ∈ L2(ℝd) up to a constant phase factor. The separation distance can be chosen independent of the degree of h and proportional to the square root of the dimension. The proof combines a discrete norming inequality, based on a multidimensional Remez inequality and VC-dimension bounds, with tail estimates for the reproducing kernel. A formalization of our main result in Lean 4 is also provided.
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BibTeX
@misc{pont2026phase,
title = {{Phase retrieval from a uniformly discrete point set}},
author = {Jaume de Dios Pont and Josef Greilhuber and Lukas Liehr and Mitchell A. Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28341},
eprint = {2609.28341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28341},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Phase retrieval from a uniformly discrete point set
AU - Jaume de Dios Pont
AU - Josef Greilhuber
AU - Lukas Liehr
AU - Mitchell A. Taylor
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28341
UR - https://arxiv.org/abs/2609.28341
AB - We prove that for every window w of the form w(x) = e^−π|x|^2 h(x), where h is a polynomial, there exists a uniformly discrete set of points 𝒮 ⊂ ℝ^2d such that the magnitude of the short-time Fourier transform V_w f on 𝒮 determines every f ∈ L^2(ℝ^d) up to a constant phase factor. The separation distance can be chosen independent of the degree of h and proportional to the square root of the dimension. The proof combines a discrete norming inequality, based on a multidimensional Remez inequality and VC-dimension bounds, with tail estimates for the reproducing kernel. A formalization of our main result in Lean 4 is also provided.
ER -