On Fourier frames
By J. Ortega‐Cerdà, Kristian Seip. 2000
133 citations recorded by the source corpus.
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 $\psi$ such that the sequence $\{\lamdda_k\}){k\in \Bbb Z}$ defined by $\lambda_k + \psi(\lambda_k)=k$ is a sampling.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/513d51127cf88454aec2c532eea5016a289cdd6d
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In Applied Mathematics
Citations in the corpus
- Beurling-Malliavin theory for Toeplitz kernels (2007, unknown): “Denote h(x) = |x|−αw(x).”
- Frames for Undergraduates (2007, background): “Thus, the question “Which subsets {λn}n∈Z ⊂ R are sets of sampling for B 1 2 ?” is equivalent to the question, “Which subsets {λn}n∈Z ⊂ R have the property that {e−2πiξλn : n ∈ Z} forms a frame for L(2)[− 12 , 1 2 ]?” This second question was initially asked by Duffin and Schaeffer in 1952 [27], and”
- Weak convergence and spectrality of infinite convolutions (2022, methodology): “In the classical case, for the normalized Lebesgue measure on a finite interval, the frame spectrum is essentially characterized by the Beurling lower uniform density [37] and the complete description of frame spectra is given by Ortega-Cerdá and Seip [45].”
- Exponential spectra in L2(μ) (2013, background): “If K is a unit interval, its F-spectrum was completely classified in terms of de Brange’s theory of entire functions [23].”
- Sampling in paley-wiener spaces on combinatorial graphs (2008, background): “The theory of irregular sampling was very active for many years [1], [2], [6], [19], [24], [23], [20], [39].”
- On Fourier frame of absolutely continuous measures (2011, methodology): “Ortega-Cerdà and Seip recently completely characterized the frame spectrum L2([0,1], dx) using de Branges’ theory of Hilbert space of entire functions [19].”