ON THE EFFECT OF ZERO-FLIPPING ON THE STABILITY OF THE PHASE RETRIEVAL PROBLEM IN THE PALEY-WIENER CLASS
PEREZ, Rolando
Institut de Mathématiques de Bordeaux [IMB]
Institute of Mathematics, University of the Philippines Diliman
Institut de Mathématiques de Bordeaux [IMB]
Institute of Mathematics, University of the Philippines Diliman
PEREZ, Rolando
Institut de Mathématiques de Bordeaux [IMB]
Institute of Mathematics, University of the Philippines Diliman
< Reduce
Institut de Mathématiques de Bordeaux [IMB]
Institute of Mathematics, University of the Philippines Diliman
Language
en
Article de revue
This item was published in
Monatshefte für Mathematik. 2022-08, vol. 198, n° 4, p. 757-776
Springer Verlag
English 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 ...Read more >
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 $b$.Read less <
English Keywords
phase retrieval
Paley-Wiener class
zero-flipping
stability
ANR Project
Noyaux reproduisants en Analyse et au-delà - ANR-18-CE40-0035
Origin
Hal imported