ON THE EFFECT OF ZERO-FLIPPING ON THE STABILITY OF THE PHASE RETRIEVAL PROBLEM IN THE PALEY-WIENER CLASS - Archive ouverte HAL Access content directly
Journal Articles Monatshefte für Mathematik Year : 2022

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 $L>0$, \textit{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 $b$.
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Dates and versions

hal-03164251 , version 1 (09-03-2021)

Identifiers

Cite

Philippe Jaming, Karim Kellay, Rolando Perez. ON THE EFFECT OF ZERO-FLIPPING ON THE STABILITY OF THE PHASE RETRIEVAL PROBLEM IN THE PALEY-WIENER CLASS. Monatshefte für Mathematik, 2022, 198 (4), pp.757-776. ⟨10.1007/s00605-022-01716-y⟩. ⟨hal-03164251⟩

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