Arithmetic progressions and holomorphic phase retrieval

TL;DR

The paper addresses the problem of recovering a holomorphic function from its modulus on sampling sets, establishing a sharp rigidity criterion: a set of the form $e^{i\theta}({\mathbb{R}}+i\Lambda)$ yields uniqueness for holomorphic phase retrieval in a broad function space ${\mathcal{F}}$ containing all exponentials if and only if the index set $\Lambda$ is not contained in an arithmetic progression. The authors leverage zero-set analysis, Weierstrass factorization, and periodicity arguments to prove necessity and sufficiency, and then translate the main result into consequences for Gabor phase retrieval across spaces like $L^2(\mathbb{R}_+)$ and the Hardy/Bargmann framework, including perturbations of lattices that restore uniqueness where lattices fail. They further derive Wright-type results for families of self-adjoint operators doing phase retrieval, offering constructive pathways to Pauli-type uniqueness results. The work unifies holomorphic phase retrieval with sampling theory and transform-domain phase retrieval, demonstrating that non-AP sampling patterns (including perturbed lattices) can yield robust uniqueness guarantees in practical settings. These findings have potential implications for designing phaseless measurement schemes in time-frequency analysis and quantum-like phase problems.

Abstract

We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ 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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:main']}