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 $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ 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 ${\mathfrak {F}}_af$ 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 $\inf _{|c|=1}\Vert cf-{\mathfrak {F}}_af\Vert _2$. We then investigate the stability of a double zero-flipping $\inf _{|c|=1}\Vert cf-{\mathfrak {F}}_{{\bar{a}}}{\mathfrak {F}}_bf\Vert _2$ which turns out to be also the stability between two zero-flipping $\inf _{|c|=1}\Vert c{\mathfrak {F}}_bf-{\mathfrak {F}}_af\Vert _2$. We show that this quantity is dominated by the distance between a and b.