A Panorama of Sampling Theory

Abstract

By a sampling function we mean a member \(\varphi \) of a vector space V of, preferably, continuous, \(\mathbf{C}\)-valued functions on a topological space X for which there is an orbit \(G \cdot {x}_{0}\) of a countable abelian group G acting continuously on X, and each f∈V is the sum of the terms \(f(k \cdot {x}_{0})\varphi (k \cdot x)\), \(k \in G\). Such a recovery formula generalizes the well-known Shannon sampling formula. This chapter presents a general discussion of sampling theory and introduces several new classes of sampling functions \(\varphi : \mathbf{R} \rightarrow \mathbf{C}\) for sampling sets of the form \(\mathbf{Z} + {x}_{0}\). In Sect.2 we discuss the very close connection between general convolution idempotents and sampling functions. In Sect.3 we review the properties of the Zak transform and use it to construct a large family of continuous sampling functions \(\varphi \in {L}^{2}(\mathbf{R})\) where \(\{{T}_{k}\varphi : k \in \mathbf{Z}\}\) is a frame for the principal shift-invariant space \({V }_{\varphi } =\langle \varphi \rangle\) generated by \(\varphi \). This family includes all band-limited sampling functions as well as all continuous sampling functions \(\varphi \in {V }_{\psi }\), \(\psi \in {C}_{c}(\mathbf{R})\). In Sect.4 we look at a class of continuous functions \(\psi \) which do not generate (via the Z-transform) any square-integrable sampling functions and use the Laurent transform (or Z-transform) to show how \(\psi \) generates a possibly infinite family of non-square-integrable sampling functions. In Sect.5 we sketch the manner in which purely algebraic tools lead to construction of a very large class of convolution idempotents and associated sampling functions that cannot be obtained by Zak or Laurent transform methods.