The Zak transform on Gelfand-Shilov and modulation spaces with applications to operator theory. (English) Zbl 1455.42032
Summary: We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.
MSC:
| 42C20 | Other transformations of harmonic type |
| 43A32 | Other transforms and operators of Fourier type |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |
| 35A22 | Transform methods (e.g., integral transforms) applied to PDEs |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Zak transform; quasi-periodic; Wiener spaces; modulation spaces; Gelfand-Shilov spaces; quasi-Banach spaces; characterizationsReferences:
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