Sampling via Generating Functions

Stephen D. Casey

Abstract:

We develop connections between some of the most powerful theories in
analysis, tying the Shannon sampling formula to Cauchy's integral and
residue formulae, Jacobi interpolation,
and Levin's sine-type functions. The techniques use tools from complex
analysis, and in particular, the Cauchy theory and the theory of entire functions,
to realize sampling sets $\Lambda$ as zero sets of well-chosen entire functions
(sampling set {\emph{generating functions}}). We then reconstruct the signal
from the set of samples using the Cauchy-Jacobi machinery.
These methods give us powerful tools for creating a variety of general
sampling formulae, e.g., allowing us to derive Shannon sampling and Papoulis
generalized sampling via Cauchy theory and sampling in radial domains.

Submission Type: Full Paper