Zak Transform and non-uniqueness in an extension of Pauli's phase retrieval problem
Language
en
Article de revue
This item was published in
Analysis Mathematica. 2016, vol. 42, p. 185-201
Springer Verlag
English Abstract
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 ...Read more >
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 $Z$.Read less <
English Keywords
Zak transform
Weyl-Heisenberg transform
Fractional Fourier Transform
Pauli problem
Phase Retrieval
ANR Project
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
Origin
Hal imported