JAMING, Philippe; KELLAY, Karim

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JAMING, Philippe
Institut de Mathématiques de Bordeaux [IMB]

KELLAY, Karim
Institut de Mathématiques de Bordeaux [IMB]

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Article de revue

Ce document a été publié dans

Journal d'analyse mathématique. 2018, vol. 134, p. 273-301

Springer

Résumé en anglais

Let $\Lambda$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $\Gamma\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(\Gamma)$ the subset of finite measures on $\Gamma$ that are absolutely continuous with respect to arc length on $\Gamma$. For such a $\mu$, $\widehat{\mu}$ denotes the Fourier transform of $\mu$. Following Hendenmalm and Montes-Rodríguez, we will say that $(\Gamma,\Lambda)$ is a Heisenberg Uniqueness Pair if $\mu\in\mathcal{A}\mathcal{C}(\Gamma)$ is such that $\widehat{\mu}=0$ on $\Lambda$, then $\mu=0$. The aim of this paper is to provide new tools to establish this property. To do so, we will reformulate the fact that $\widehat{\mu}$ vanishes on $\Lambda$ in terms of an invariance property of $\mu$ induced by $\Lambda$. This leads us to a dynamical system on $\Gamma$ generated by $\Lambda$. The investigation of this dynamical system allows us to establish that $(\Gamma,\Lambda)$ is a Heisenberg Uniqueness Pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also allows to have a better geometric intuition on why $(\Gamma,\Lambda)$ is a Heisenberg Uniqueness Pair.< Réduire

Mots clés en anglais

Uncertainty principles

annihilating pairs

Heisenberg pairs

URI

https://oskar-bordeaux.fr/handle/20.500.12278/194728

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