Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transforms
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | LE MANACH, Florian | |
| dc.date.accessioned | 2024-04-04T03:06:57Z | |
| dc.date.available | 2024-04-04T03:06:57Z | |
| dc.date.created | 2017-07-01 | |
| dc.date.issued | 2018-06-01 | |
| dc.identifier.issn | 0022-247X | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193385 | |
| dc.description.abstractEn | We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{Z})$. It is known that a vector $u$ is cyclic in $\ell^2(\mathbb{Z})$ if and only if the zero set, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, has Lebesgue measure zero and $\log |\widehat{u}| \not \in L^1(\mathbb{T})$, where $\mathbb{T}$ is the unit circle. Here we show that, unlike $\ell^2(\mathbb{Z})$, there is no characterization of the cyclicity of $u$ in $\ell^p(\mathbb{Z})$, $1<p<2$, in terms of $\mathcal{Z}(\widehat{u})$ and the divergence of the integral $\int_\mathbb{T} \log |\widehat{u}| $. Moreover we give both necessary conditions and sufficient conditions for $u$ to be cyclic in $\ell^p(\mathbb{Z})$, $1<p<2$. | |
| dc.language.iso | en | |
| dc.publisher | Elsevier | |
| dc.subject.en | weighted $\ell^p$ spaces | |
| dc.subject.en | cyclicity | |
| dc.title.en | Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transforms | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1016/j.jmaa.2017.12.057 | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.subject.hal | Mathématiques [math]/Analyse classique [math.CA] | |
| dc.identifier.arxiv | 1707.09773 | |
| bordeaux.journal | Journal of Mathematical Analysis and Applications | |
| bordeaux.page | 967-981 | |
| bordeaux.volume | 462 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 1 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01570349 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01570349v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Mathematical%20Analysis%20and%20Applications&rft.date=2018-06-01&rft.volume=462&rft.issue=1&rft.spage=967-981&rft.epage=967-981&rft.eissn=0022-247X&rft.issn=0022-247X&rft.au=LE%20MANACH,%20Florian&rft.genre=article |
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