Generalizations of Shannon Sampling Theory

Sampling Theory is the branch of mathematics that seeks to reconstruct functions from knowledge of their values at a discrete set of points. It is one of the most important techniques used in communication engineering and information theory. The fundamental result in Sampling Theory, most commonly known as Shannon's Sampling Theorem, is explained on R and ${\bf R\sp {n}}.$ We give proofs and simulations of the theorem in both one and several variables. We then show how results in multichannel deconvolution by Berenstein, Yger, Casey, Walnut, et al. can be used to reconstruct a function of one variable by sampling at non-commensurate rates. The benefit of this technique is that the individual sampling rates are much lower than the sampling rate of the traditional sampling formula. Computer simulations generated using Mathcad are included.