A Note on the Phase Retrieval of Holomorphic Functions

TL;DR

This paper investigates phase retrieval problems for holomorphic functions, proving uniqueness results under specific geometric and boundary conditions, including cases involving the Nevanlinna class.

Contribution

It establishes new uniqueness theorems for holomorphic functions based on modulus data on intersecting segments and boundary circles, extending phase retrieval theory.

Findings

Uniqueness of holomorphic functions up to unimodular constants from modulus on intersecting segments with irrational angles.

Uniqueness of functions in the Nevanlinna class from modulus on the unit circle and an interior circle.

Extension of phase retrieval results to broader classes of holomorphic functions.

Abstract

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi$. We also prove that if f and g are functions in the Nevanlinna class, and if |f | = |g| on the unit circle and on a circle inside the unit disc, then f = g up to the multiplication of a unimodular constant.