Type: Article
Publication Date: 2002-05-01
Citations: 90
DOI: https://doi.org/10.2307/3062132
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 {λ k } k∈Z defined by λ k + ψ(λ k ) = k is sampling.