Approximating Paley-Wiener Functions by Smoothed Step Functions Original Research Article

A function ƒ with compactly supported Fourier transform can be approximated by a step function a which coincides with ƒ at regularly spaced points sk, k ∈ Z. For suitable s, the functions ƒ and a have the same L2 norm. By modifying a so that its Fourier transform shares the same compact support as that of ƒ, an analytic function is obtained which approximates ƒ, the accuracy depending on s.