On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class

Abstract

In the classical phase retrieval problem in the Paley-Wiener class $P{W}_L$ for $L>0$, i.e. to recover $f\in P{W}_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by ${\mathfrak{F}}_{a}f$ the resulting function. The operator ${\mathfrak{F}}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of ${\mathfrak{F}}_a$. We show that ${\mathfrak{F}}_{a}f$ is no longer bandlimited but is still wide-banded. We then investigate the effect of ${\mathfrak{F}}_a$ on the stability of phase retrieval by estimating the quantity ${\inf }_{|c|=1}\Vert cf-{\mathfrak{F}}_{a}f{\Vert }_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity ${\inf }_{|c|=1}\Vert c{\mathfrak{F}}_{b}f-{\mathfrak{F}}_{a}f{\Vert }_2$. We show that this quantity is dominated by the distance between $a$ and $b$.