On a dense set of functions determined by sampled Gabor magnitude

TL;DR

This paper demonstrates that functions with Bargmann transforms as entire functions of exponential type can be uniquely recovered from sampled Gabor magnitude data on dense sets, advancing phase retrieval understanding.

Contribution

It characterizes conditions under which such functions can be uniquely reconstructed from magnitude-only samples, expanding phase retrieval theory for specific function classes.

Findings

Unique recovery of entire functions of exponential type from sampled magnitude data.

Sampling on dense shifted lattices suffices for phase retrieval of these functions.

The results specify lower Beurling density requirements for uniqueness.

Abstract

We study the problem of recovering a function from the magnitude of its Gabor transform sampled on a discrete set. While it is known that uniqueness fails for general square integrable functions, we show that phase retrieval is possible for a dense class of signals: specifically, those whose Bargmann transforms are entire functions of exponential type. Our main result characterises when such functions can be uniquely recovered (up to a global phase) from magnitude only data sampled on uniformly discrete sets of sufficient lower Beurling density. In particular, we prove that every entire function of exponential type is uniquely determined (up to a global phase) among all second order entire functions by its modulus on a sufficiently dense shifted lattice with suitable structure.