Phase Retrieval and Magnitude Retrieval of Entire Functions

This article concerns the problem of determining an entire function f when |f (x)| is known for every real x. For functions of finite order it is shown that |f (x)| and |f (x + b) – |f (x)| determine f up to a factor with period b and the involution f(z) \rightarrow \overbar{f (\overbar{z})}. Applications are given in the cases where f is a polynomial, a trigonometric polynomial, or where it corresponds to a band-limited signal. Similar results are obtained for the analogous problem of determining f when (arg(f (x)) is known.