The Balian-Low theorem for locally compact abelian groups and vector bundles
| العنوان: | The Balian-Low theorem for locally compact abelian groups and vector bundles |
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| المؤلفون: | Ulrik Enstad |
| سنة النشر: | 2019 |
| مصطلحات موضوعية: | Applied Mathematics, General Mathematics, 010102 general mathematics, Zak transform, Mathematics - Operator Algebras, Second-countable space, Vector bundle, 010103 numerical & computational mathematics, 01 natural sciences, Functional Analysis (math.FA), Combinatorics, Annihilator, Mathematics - Functional Analysis, Compact space, Balian–Low theorem, FOS: Mathematics, Locally compact space, 0101 mathematics, Abelian group, 42C15 43A70 46L08, Operator Algebras (math.OA), Mathematics |
| الوصف: | Let $\Lambda$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $\Lambda^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $\eta$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_{\tau} T_{\lambda} \eta \}_{\lambda \in \Lambda, \tau \in \Lambda^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $\Lambda = \alpha \mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\Lambda)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\Lambda) \times (\widehat{G}/\Lambda^{\perp})$. We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert $C^*$-modules. As an application, we prove a new Balian-Low theorem for the group $\mathbb{R} \times \mathbb{Q}_p$, where $\mathbb{Q}_p$ denotes the $p$-adic numbers. Comment: 29 pages |
| اللغة: | English |
| تدمد: | 0021-7824 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::75ef9e477907dad56ddd6fe634136e27 http://arxiv.org/abs/1905.06827 |
| Rights: | OPEN |
| رقم الانضمام: | edsair.doi.dedup.....75ef9e477907dad56ddd6fe634136e27 |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 00217824 |
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