Frames Generated by Actions of Locally Compact Groups
| dc.contributor.advisor | Bownik, Marcin | |
| dc.contributor.author | Iverson, Joseph | |
| dc.date.accessioned | 2016-10-27T18:36:03Z | |
| dc.date.available | 2016-10-27T18:36:03Z | |
| dc.date.issued | 2016-10-27 | |
| dc.description.abstract | Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ \rho(x) f_j \colon x \in G, j \in I\}$ for families $\{f_j\}_{j \in I}$ in $\mathcal{H}_\rho$. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in $\mathcal{H}_\rho$ to form a continuous frame. We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group $\Gamma \supset G$ containing $G$ as a closed subgroup, and we let $\rho$ be the action of $G$ on $L^2(\Gamma)$ by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on $L^2(\Gamma)$ which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of $L^2(\Gamma)$ which are invariant under left translation by $G$. The two projects give compatible outcomes. This dissertation contains previously published material. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/20443 | |
| dc.language.iso | en_US | |
| dc.publisher | University of Oregon | |
| dc.rights | All Rights Reserved. | |
| dc.subject | Compact group | en_US |
| dc.subject | Dual integrable | en_US |
| dc.subject | Group frame | en_US |
| dc.subject | LCA group | en_US |
| dc.subject | Shift-invariant | en_US |
| dc.subject | Zak transform | en_US |
| dc.title | Frames Generated by Actions of Locally Compact Groups | |
| dc.type | Electronic Thesis or Dissertation | |
| thesis.degree.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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