2015/01/16 by Simon Andréys, Simon Andreys, Philippe Jaming +2 · 1 citation
Mathematics · Physics and Astronomy · #Advanced X-ray Imaging Techniques #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #math-ph #math.CA #math.CV #math.FA #math.MP
paper · pdf · doi:10.48550/arxiv.1501.03905
arxiv created 2015/01/16 · openalex publication_date 2015/01/16 · arxiv updated 2015/01/19 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform \ff_α started by the second author. We here extend a method of A.E.J.M Janssen to show that there is a countable set \qq such that for every finite subset å⊂ \qq, there exist two functions f,g not multiple of one an other such that |\ff_αf|=|\ff_αg| for every α∈ å. Equivalently, in quantum mechanics, this result reformulates as follows: if Q_α=Qcosα+Psinα (Q,P be the position and momentum observables), then \Q_α,α∈å\ is not informationally complete with respect to pure states. This is done by constructing two functions \ffi,ψ such that \ff_α\ffi and \ff_αψ have disjoint support for each α∈ å. To do so, we establish a link between \ff_α[f], α∈ \qq and the Zak transform Z[f] generalizing the well known marginal properties of Z.