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

Matthias Wellershoff

TL;DR

This work characterises when the magnitudes of finite-order entire functions determine the function up to a global phase from data on lines. It distinguishes intersecting and parallel cases, proving uniqueness for irrational intersection angles and providing structured factorizations for rational angles; it then extends to multiple parallel lines, including three-line and equidistant parallel-line settings, and constructs universal counterexamples. The results are tied to Gabor phase retrieval through the Bargmann/Fock-space correspondence, yielding practical implications for time-frequency analysis and sampling. Overall, the paper advances the theory of phase retrieval for entire functions and offers concrete symbolic forms for the recovered functions under line-based magnitude data with explicit conditions on zeros and exponential factors.

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.

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

TL;DR

This work characterises when the magnitudes of finite-order entire functions determine the function up to a global phase from data on lines. It distinguishes intersecting and parallel cases, proving uniqueness for irrational intersection angles and providing structured factorizations for rational angles; it then extends to multiple parallel lines, including three-line and equidistant parallel-line settings, and constructs universal counterexamples. The results are tied to Gabor phase retrieval through the Bargmann/Fock-space correspondence, yielding practical implications for time-frequency analysis and sampling. Overall, the paper advances the theory of phase retrieval for entire functions and offers concrete symbolic forms for the recovered functions under line-based magnitude data with explicit conditions on zeros and exponential factors.

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 does not uniquely determine it up to global phase, for all positive integers . Our results have direct implications for Gabor phase retrieval.
Paper Structure (17 sections, 17 theorems, 90 equations, 2 figures)

This paper contains 17 sections, 17 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation, preliminaries and overview of existing literature
  3. Notation
  4. Preliminaries
  5. Background and overview of existing literature
  6. Characterisation of finite order entire functions with common magnitudes on two lines
  7. Intersecting lines
  8. Parallel lines
  9. Further results
  10. Uniqueness for three parallel lines
  11. Characterisation of finite order entire functions with common magnitudes on equidistant parallel lines
  12. Construction of universal counterexamples
  13. A remark on Gabor transform phase retrieval
  14. Expansion of a function as an infinite product
  15. Characterisation of finite order entire functions with common magnitudes on equidistant parallel lines
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\ell_1,\ell_2,\ell_3 \subset \mathbb{C}$ be three parallel lines. Let $a > 0$ denote the distance between $\ell_1$ and $\ell_2$ and let $b > 0$ denote the distance between $\ell_2$ and $\ell_3$. Assume that $\tfrac{b}{a}$ is irrational and let $f$ as well as $g$ be entire functions. Then, $\lve

Figures (2)

  • Figure 1: The grouping of zeroes in $\mathop{\mathrm{supp}}\nolimits(M_\mathrm{d})$ using the rotational symmetry with angle $2 \theta$. The dotted line indicates the open boundary of $C_\theta$. Note that, for all $a,a' \in \mathop{\mathrm{supp}}\nolimits(M_\mathrm{d})$, it holds that $\rho_{2 \theta}^{k} a, \rho_{2 \theta}^{k} a' \in \mathop{\mathrm{supp}}\nolimits(M_\mathrm{d})$, for $k=1,\dots,n$.
  • Figure 2: Depiction of $S_{y_0}$. Note that the dashed line indicates that $\mathbb{R} - \mathrm{i} y_0$ is not contained in $S_{y_0}$.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof : Proof of Lemma \ref{['lem:folklore']}
  • Theorem 4
  • Remark 5
  • Theorem 6: jaming2014uniqueness
  • Remark 7
  • Theorem 8
  • proof
  • ...and 28 more