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

In this paper, we consider the full or partial recovery of entire functions from magnitude measurements on different subsets of the complex plane. This so-called phase retrieval for entire functions is inspired by its manifold connections to other phase retrieval problems. A particular connection, illuminated in more detail in this paper, is that to Gabor phase retrieval which can be made by using the Bargmann transform and the Fock space. By applying well-known techniques from complex analysis - in particular, the famous Hadamard factorisation theorem - we develop many known and numerous new results for the phase retrieval of entire functions. Among other things, we provide a full classification of all (finite order) entire functions whose magnitudes agree on two arbitrary lines in the complex plane as well as a full classification of all entire functions of exponential type whose magnitudes agree on infinitely many equidistant parallel lines. Our results have interesting implications for Gabor phase retrieval such as giving a full classification of all signals whose Gabor magnitudes agree on two arbitrary lines in the time-frequency plane; or, yielding a machinery with which to generate signals whose Gabor magnitudes agree on infinitely many equidistant lines. The latter has already been harnessed to propose certain counterexamples to sampled Gabor phase retrieval in the literature. We show here that one may also apply this machinery to generate so-called ``universal counterexamples'': signals which cannot be recovered (up to global phase) from magnitude measurements of their Gabor transforms on infinitely many equidistant parallel lines - no matter how small the distance between those lines