Shannon's sampling theorem in a distributional setting

TL;DR

This paper extends Shannon's sampling theorem to cases where the Fourier transform is a compactly supported distribution, broadening the theorem's applicability beyond functions in L^2(R).

Contribution

It generalizes the classical sampling theorem to include Fourier transforms that are compactly supported distributions, not just functions.

Findings

Sampling formula holds for compactly supported distributions

Generalization includes broader class of Fourier transforms

Maintains exact reconstruction from samples

Abstract

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-\pi,\pi) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(\pi (t -n)))/(\pi (t-n)) (t in R). In this article we prove a generalization of this result under the assumption that F is a compactly supported distribution with its support contained in (-\pi,\pi).