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 $\inf_{ |c|=1}\|cf-\mathfrak{ F}_af\|_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity $\inf_{ |c|=1}\|c\mathfrak{ F}_bf-\mathfrak{ F}_af\|_2$. We show that this quantity is dominated by the distance between $a$ and $b$.
Categories: math.CA math.CV math.FA