Phase retrieval of entire functions and its implications for Gabor phase retrieval

TL;DR

This paper characterizes entire functions based on their magnitudes on specific lines in the complex plane, with implications for phase retrieval in Gabor analysis, revealing conditions for unique determination of functions up to global phase.

Contribution

It provides new characterizations of entire functions from magnitude data on lines, extending to cases with multiple lines and rational independence, impacting Gabor phase retrieval.

Findings

Magnitudes on two arbitrary lines determine finite order entire functions.

Magnitudes on infinitely many equidistant parallel lines determine second order entire functions.

Magnitudes on three rationally independent lines uniquely determine the entire function up to global phase.

Abstract

We characterise all pairs of finite order entire functions whose magnitudes agree on two arbitrary lines in the complex plane by means of the Hadamard factorisation theorem. Building on this, we also characterise all pairs of second order entire functions whose magnitudes agree on infinitely many equidistant parallel lines. Furthermore, we show that the magnitude of an entire function on three parallel lines, whose distances are rationally independent, uniquely determines the function up to global phase, and that there exists a first order entire function whose magnitude on the lines $\mathbb{R} + \tfrac{\mathrm{i}}{n} \mathbb{Z}$ does not uniquely determine it up to global phase, for all positive integers $n$. Our results have direct implications for Gabor phase retrieval.