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 WL for L > 0, i.e. to recover f ∈ P WL 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 Faf the resulting function. The operator Fa 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 Fa. We show that Faf is no longer bandlimited but is still wide-banded. We then investigate the effect of Fa on the stability of phase retrieval by estimating the quantity inf |c|=1 cf − Faf 2 . We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity inf |c|=1 cF b f − Faf 2. We show that this quantity is dominated by the distance between a and b.