A dynamical system approach to Heisenberg Uniqueness Pairs
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | JAMING, Philippe | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | KELLAY, Karim | |
| dc.date.accessioned | 2024-04-04T03:22:09Z | |
| dc.date.available | 2024-04-04T03:22:09Z | |
| dc.date.issued | 2018 | |
| dc.identifier.issn | 0021-7670 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194728 | |
| dc.description.abstractEn | 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. | |
| dc.description.sponsorship | 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 | |
| dc.language.iso | en | |
| dc.publisher | Springer | |
| dc.subject.en | Uncertainty principles | |
| dc.subject.en | annihilating pairs | |
| dc.subject.en | Heisenberg pairs | |
| dc.title.en | A dynamical system approach to Heisenberg Uniqueness Pairs | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Analyse classique [math.CA] | |
| dc.identifier.arxiv | 1312.6236 | |
| bordeaux.journal | Journal d'analyse mathématique | |
| bordeaux.page | 273-301 | |
| bordeaux.volume | 134 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00921685 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00921685v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20d'analyse%20math%C3%A9matique&rft.date=2018&rft.volume=134&rft.spage=273-301&rft.epage=273-301&rft.eissn=0021-7670&rft.issn=0021-7670&rft.au=JAMING,%20Philippe&KELLAY,%20Karim&rft.genre=article |
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