Arithmetic progressions and holomorphic phase retrieval

TL;DR

This paper characterizes when a holomorphic function can be uniquely determined by its absolute value on certain sets, revealing connections to arithmetic progressions and applications in phase retrieval problems.

Contribution

It provides a new criterion based on arithmetic progressions for the uniqueness of holomorphic functions from magnitude data, impacting phase retrieval and related fields.

Findings

Uniqueness of holomorphic functions is characterized by the non-inclusion of certain sets in arithmetic progressions.

Sets like Z x Z̃ serve as uniqueness sets for Gabor phase retrieval under specific conditions.

The results have implications for phase retrieval problems in signal processing and quantum mechanics.

Abstract

We study the determination of a holomorphic function from its absolute value. Given a parameter $\theta \in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $\Lambda \subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{\xi z}, \, \xi \in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{i\theta}(\mathbb{R} + i\Lambda)\}$ if and only if $\Lambda$ is not contained in an arithmetic progression $a\mathbb{Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $\mathbb{Z} \times \tilde{\mathbb{Z}}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2(\mathbb{R}_+)$, provided that $\tilde{\mathbb{Z}}$ is a suitable perturbation of the integers.