ANDREYS, Simon; JAMING, Philippe

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ANDREYS, Simon
Institut de Mathématiques de Bordeaux [IMB]

JAMING, Philippe
Institut de Mathématiques de Bordeaux [IMB]

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Article de revue

Ce document a été publié dans

Analysis Mathematica. 2016, vol. 42, p. 185-201

Springer Verlag

Résumé en anglais

The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform $\ff_\alpha$started by the second author.We here extend a method of A.E.J.M Janssen toshow that there is a countable set $\qq$ such that for every finite subset $\aa\subset \qq$, there exist twofunctions $f,g$ not multiple of one an other such that $|\ff_\alpha f|=|\ff_\alpha g|$ for every $\alpha\in \aa$.Equivalently, in quantum mechanics, this result reformulates as follows:if $Q_\alpha=Q\cos\alpha+P\sin\alpha$ ($Q,P$ be the position and momentum observables),then $\{Q_\alpha,\alpha\in\aa\}$ is not informationally complete with respect to pure states.This is done by constructing two functions $\ffi,\psi$ such that $\ff_\alpha\ffi$ and $\ff_\alpha\psi$ have disjoint support for each $\alpha\in \aa$. To do so, we establish a link between $\ff_\alpha[f]$, $\alpha\in \qq$ and the Zak transform $Z[f]$generalizing the well known marginal properties of $Z$.< Réduire

Mots clés en anglais

Zak transform

Weyl-Heisenberg transform

Fractional Fourier Transform

Pauli problem

Phase Retrieval

URI

https://oskar-bordeaux.fr/handle/20.500.12278/194510

Project ANR

Conséquences à long terme de l'exposition péripubertaire aux cannabinoides: étude comportementale et transcriptionnelle chez le rat et analyse moléculaire chez l'homme - ANR-06-NEUR-0044

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251