Phase retrieval in Fock space and perturbation of Liouville sets

TL;DR

This paper demonstrates that phase retrieval in Fock space is solvable using irregular, perturbed Liouville sets, establishing new conditions for uniqueness sets with finite density, and extends results to Gabor phase retrieval in real-valued function subspaces.

Contribution

It introduces a novel approach using perturbations of Liouville sets to achieve solvability of phase retrieval in Fock space, including the first construction of finite density uniqueness sets.

Findings

Solvability of phase retrieval with irregular sampling sets.

Liouville sets are intermediate between stable sampling and uniqueness sets.

Reduced size of uniqueness sets for real-valued and even functions.

Abstract

We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this research field asserts that phaseless sampling on lattices of arbitrary density renders the problem unsolvable. The present study establishes solvability when using irregular sampling sets of the form $A \cup B \cup C$, where $A, B,$ and $C$ constitute perturbations of a Liouville set, i.e., a set with the property that all functions in Fock space bounded on the set are constant. The sets $A, B,$ and $C$ adhere to specific geometrical conditions of closeness and noncollinearity. We show that these conditions are sufficiently generic so as to allow the perturbations to be chosen also at random. By proving that Liouville sets occupy an intermediate position between sets of stable sampling and sets of uniqueness, we obtain the first construction of uniqueness sets for the phase retrieval problem in Fock space having a finite density. The established results apply to the Gabor phase retrieval problem in subspaces of $L^2(\mathbb{R})$, where we derive additional reductions of the size of uniqueness sets: for the class of real-valued functions, uniqueness is achieved from two perturbed lattices; for the class of even real-valued functions, a single perturbation suffices, resulting in a separated set.