Sampling theorems and the contour integral method

Abstract

Various forms of the Whittaker-Kotelnikov -Shannon sampling theorem, which allow certain entire functions to be represented by interpolatory series, are derived by viewing them as sums of residues. Series in which the sample points are distributed in one and in two dimensions are considered.This contour integral method provides a powerful way of generating the correct form of the series, and of obtaining the uniform convergence.Particular emphasis is placed upon series representations involving derivative samples as well as samples of the function itself. The general derivative sampling series is treated, as well as derivative sampling at points which are slightly perturbed from uniform spacing. Finally, derivative sampling at lattice points in the complex plane is considered

AMS(MOS):

• 42A38

Keywords:

• Paley-Wiener functions
• Kotelnikov-Shannon sampling
• contour integration