Analytic approaches to the phase retrieval problem

Given a Banach function space X and an operator T on X, the phase retrieval problem concerns the question of whether every f in X is determined (up to a multiplicative factor) by |Tf|, i.e., the modulus of Tf. Problems of this nature were first studied by Pauli, who asked whether a square-integrable function is determined by its modulus and the modulus of its Fourier transform. Since then, the problem has been investigated from multiple directions. I will give an overview of important results in this research field and outline recent developments. The focus of the talk will be on approaches to the phase retrieval problem that draw from techniques in Complex, Harmonic, and Functional Analysis.