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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The discrete nature of the Paley-Wiener spaces
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by Carolyn Eoff PDF
Proc. Amer. Math. Soc. 123 (1995), 505-512 Request permission

Abstract:

The Shannon Sampling Theorem suggests that a function with bandwidth $\pi$ is in some way determined by its samples at the integers. In this work we make this idea precise for the functions in the Paley-Wiener space ${E^p}$. For $p > 1$, we make a modest contribution, but the basic result is implicit in the classical work of Plancherel and Pólya (1937). For $0 < p \leq 1$, we combine old and new results to arrive at a characterization of ${E^p}$ via the discrete Hilbert transform. This indicates that for such entire functions to belong to ${L_p}({\mathbf {R}},dx)$, not only is a certain rate of decay required, but also a certain subtle oscillation.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 505-512
  • MSC: Primary 42A38; Secondary 42A65, 94A12
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1219724-X
  • MathSciNet review: 1219724