Fourier frames
We solve the problem of Duffin and Schaeffer (1952) of characterizing those sequences of real frequencies which generate Fourier frames. Equivalently, we characterize the sampling sequences for the Paley-Wiener space. The key step is to connect the problem with de Branges' theory of Hilbert spa...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2002 |
| País: | España |
| Recursos: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositório: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/164422 |
| Acesso em linha: | https://hdl.handle.net/2445/164422 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Anàlisi harmònica Funcions de variables complexes Funcions analítiques Anàlisi funcional Harmonic analysis Functions of complex variables Analytic functions Functional analysis |
| Resumo: | We solve the problem of Duffin and Schaeffer (1952) of characterizing those sequences of real frequencies which generate Fourier frames. Equivalently, we characterize the sampling sequences for the Paley-Wiener space. The key step is to connect the problem with de Branges' theory of Hilbert spaces of entire functions. We show that our description of sampling sequences permits us to obtain a classical inequality of H.~Landau as a consequence of Pavlov's description of Riesz bases of complex exponentials and the John-Nirenberg theorem. Finally, we discuss how to transform our description into a working condition by relating it to an approximation problem for subharmonic functions. By this approach, we determine the critical growth rate of a non-decreasing function $\psi$ such that the sequence $\{\lambda_k\}_{k\in\Z}$ defined by $\lambda_k+\psi(\lambda_k)=k$ is sampling. |
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