The phase retrieval problem for solutions of the Helmholtz equation

Abstract

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+{\lambda }^{2}u=0$ on domains $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$. In dimension $d=2$, if $u,v$ are two such solutions then $|u|=|v|$ implies that either $u=cv$ or $u=c\bar{v}$ for some $c\in \mathbb{C}$ with $|c|=1$. In dimension $d\ge 3$, the same conclusion holds under some restriction on $u$ and $v$: either they are real valued or zonal functions or have non vanishing mean.

References in corpus (1)

• Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order