The phase retrieval problem

In the phase retrieval problem one seeks to recover an unknown function g(t) from the amplitude mod g(k) mod of its Fourier transform. Since phase and amplitude are, in general, independent of each other, it is necessary to make use of other kinds of information which implicitly or explicitly constrain the admissible solutions g(t). In this paper we survey a variety of results explaining circumstances under which g(t) may be uniquely recovered from mod g(k) mod and supplementary information. A number of explicit formulae for the phase are discussed. We pay particular attention to the phase retrieval problem as it arises in certain inverse-scattering applications.