Abstract
For a fixed singular Borel probability measure \(\mu \) on \((-1/2, 1/2)\), we give several characterizations of when an entire function is the Fourier transform of some \(f \in L^2(\mu )\). The first characterization is given in terms of criteria for sampling functions of the form \(\hat{f}\) when \(f \in L^2(\mu )\). The second characterization is given in terms of criteria for interpolation of bounded sequences on \({\mathbb {N}}_{0}\) by \(\hat{f}\). Both characterizations use the construction of Fourier series for \(f \in L^2(\mu )\) demonstrated by Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of \(\mu \).
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Acknowledgements
We thank the anonymous referees for numerous suggestions that improved the presentation of this paper. We also thank the referees for bringing to our attention the folklore result Theorem B and the proof presented there. The results of this paper were inspired while the author attended the workshop “Hilbert Spaces of Entire Functions and their Applications” at the Institute of Mathematics, Polish Academy of Sciences (IM PAN). The author thanks the organizers of the workshop for the invitation to participate and IM PAN for their hospitality.
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Communicated by Yura Lyubarskii.
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Weber, E.S. A Paley–Wiener Type Theorem for Singular Measures on \((-1/2, 1/2)\). J Fourier Anal Appl 25, 2492–2502 (2019). https://doi.org/10.1007/s00041-019-09671-3
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DOI: https://doi.org/10.1007/s00041-019-09671-3