Phase retrieval in spaces of analytic functions on the unit disk

We consider the problem of reconstructing functions f which are analytic in the open unit disk D from amplitude measurements |f| on the boundary of D and inside of D. It is shown that if f has no singular part then f is uniquely determined by its amplitude on the unit circle and by its amplitude on a circle inside the unit disk. As a special case, the second part of the paper is concerned with the phase retrieval problem for polynomials of a fixed finite degree. Then our approach yields a simple proof of the known fact that in an N-dimensional complex vector space 4N - 4 measurements are sufficient to recover every signal from its amplitude measurements. Moreover, a natural construction of corresponding measurement vectors is achieved.