The discrete nature of the Paley-Wiener spaces

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.