IJPAM: Volume 45, No. 2 (2008)

A NOTE ON PHASE RETRIEVAL OF $H^2$-FUNCTIONS

John N. McDonald
Department of Mathematics
Arizona State University
Tempe, AZ 85287-1804, USA
e-mail: mcdonald@math.la.asu.edu


Abstract.This note concerns the problem of determining an $H^2$-function $g$ when $\vert g(x)\vert$ is known for almost all $x.$ It is shown that the conditions $(\ast)\quad \vert g(x)\vert=\vert f(x)\vert$ and $(\ast\ast)\quad\vert g(qx)-g(x)\vert=\vert f(qx)-f(x)\vert,0<q<1$ together imply that either $g=Vf$ or $g=V\bar f,$ where $V$ satisfies $V(qx)=V(x).$ Under certain restrictions on $f$ and $g$ the function $V$ reduces to a constant.

Received: November 25, 2007

AMS Subject Classification: 94A12, 30D20

Key Words and Phrases: phase retrieval, Hardy space of the upper half-plane

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 45
Issue: 2