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The discrete nature of the Paley-Wiener spaces


Author: Carolyn Eoff
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
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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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DOI: https://doi.org/10.1090/S0002-9939-1995-1219724-X
Article copyright: © Copyright 1995 American Mathematical Society

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