Phase retrieval of H2-functions

This paper concerns the problem of determining an H2-function f when |f (x)| is known for almost all x. It is shown that the conditions (∗) |g(x)| = |f (x)| and (∗∗) |g(x + b) − g(x)| = |f (x + b) − f (x)| together imply that either g = Vf or g = V f¯, where V has period b. Under certain restrictions on f and g the function V takes the form V = eı(αx+β)B1B¯2, where B1, and B2, are Blaschke products. If the conditions (∗) and (∗∗) hold for three incommensurable b’s, then g is constant multiple of f . Analogous results for the problem of magnitude retrieval of H2-functions are discussed briefly.  2005 Elsevier Inc. All rights reserved.