Exponential bases, Paley-Wiener spaces and applications. (English) Zbl 1306.42048
Let \(\Omega \subset {\mathbb R}^d\) \((d\geq 1)\) be a Borel set of finite measure. The Paley-Wiener space \(\mathrm{PW}_{\Omega}({\mathbb R}^d)\) contains all functions \(f\in L^2({\mathbb R}^d)\) whose Fourier transforms are supported in \(\Omega\). For a given countable set \(A\subset {\mathbb R}^d\) and \(\Phi\in\mathrm{PW}_{\Omega}({\mathbb R}^d)\), the authors investigate the connection between the translation basis \(\{\Phi(\cdot -a);\, a \in A\}\) and the exponential basis \(\{e_a;\, a\in A\}\) with \(e_a(x) = \exp(-2\pi i\,\langle x,a\rangle)\) \((x\in \Omega)\). Roughly spoken, it is shown that \(\{\Phi(\cdot -a);\, a \in A\}\) is an orthonormal basis, a Riesz basis, resp. a frame for \(\mathrm{PW}_{\Omega}({\mathbb R}^d)\) if and only if \(\{e_a;\, a\in A\}\) is an orthonormal basis, a Riesz basis, resp. a frame for \(L^2(\Omega)\). For the case of an orthonormal basis, see B. Fuglede [J. Funct. Anal. 16, 101–121 (1974; Zbl 0279.47014)]. Later this result is extended to vector-valued Gabor systems.
Reviewer: Manfred Tasche (Rostock)
MSC:
| 42C15 | General harmonic expansions, frames |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42B35 | Function spaces arising in harmonic analysis |
Keywords:
Paley-Wiener space; orthonormal basis; Riesz basis; frame; translation basis; exponential basis; vector-valued Gabor systemCitations:
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