The Mathematics of Phase Retrieval.

The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astronomy, radar, speech recognition, quantum mechanics and, perhaps most prominently, diffraction imaging. The mathematical study of phase retrieval problems possesses a long history with a number of beautiful and deep results drawing from different mathematical fields, such as harmonic analyis, complex analysis, or Riemannian geometry. The present paper aims to present a summary of some of these results with an emphasis on recent activities. In particular we aim to summarize our current understanding of uniqueness and stability properties of phase retrieval problems.