Uniqueness results in an extension of Pauli's phase retrieval problem

JAMING, Philippe

doi:10.1016/j.acha.2014.01.003

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JAMING, Philippe
Institut de Mathématiques de Bordeaux [IMB]
Mathématiques - Analyse, Probabilités, Modélisation - Orléans [MAPMO]

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Article de revue

Ce document a été publié dans

Applied and Computational Harmonic Analysis. 2014, vol. 37, p. 413-441

Elsevier

Résumé en anglais

In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $\alpha$ have same modulus $|F_\alpha u|=|F_\alpha v|$ for some set $\tau$ of $\alpha$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $\tau$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set $\tau$ may even be reduced to a single point ({\it i.e.} one fractional Fourier transform may suffice for uniqueness in the problem).< Réduire

Mots clés en anglais

Phase retrieval

Pauli problem

Fractional Fourier transform

entire function of finite order

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190119

DOI

http://dx.doi.org/10.1016/j.acha.2014.01.003

Project ANR

Analyse Harmonique et Problèmes Inverses - ANR-07-BLAN-0247

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251

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