Year: 2000
Type: preprint
Abstract: 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 nondecreasing function $ψ$ such that the sequence $\{\lamdda_k\}){k\in \Bbb Z}$ defined by $λ_k + ψ(λ_k)=k$ is a sampling.
Source: arXiv (Cornell University)
Authors: Ortega-Cerda, Joaquim, Seip, Kristian
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4394649674