Integral transforms and sampling theorems
by
J.R. Higgins
Prof. Emeritus, Anglia Polytechnic University, Cambridge, England.

A sampling theorem asserts that members of a certain class of functions defined on a given domain can be represented in terms of their values, or samples, taken at a countable subset of the domain. There representations usually take the form of a series expansion. The importance of sampling theorems lies in the fact that they show how certain functions can be determined in their entirety by just countably many items of data.

The purpose of this lecture is to show how specific sampling theorems can be derived from a general sampling principle for functions belonging to a Hilbert space with reproducing kernel. This principle is in turn based on a theory of linear transformations of Hilbert spaces due to S. Saitoh.

Integral transforms feature largely in the search for concrete realizations of this general sampling principle. A central requirement is the presence of a sequence of points in the domain of the function which must have the property that when one argument of the kernel of an integral transform is evaluated successively at the points of the sequence, the result is a set of basis elements for an underlying Hilbert space.

There are several main avenues of approach to such a construction; here we describe three of them.

Our first approach requires the presence of a suitable eigenvalue problem with discrete eigenvalues. The kernel we are seeking arises as a general solution of the eigenvalue problem, the eigenvalues determine the sample points and the basis functions are the eigenfunctions. The interest in this approach lies mainly in the fact that a sampling series can be associated with an eigenvalue problem alongside the standard eigenfunction expansion.

For a second approach, we start with a known basis for some Hilbert space, and from it construct a kernel and an associated set of sample points which yield the given basis, as described above. The interest in this approach is that, when successful, a sampling series can be associated with a basis having some particular interest.

Taking rather the opposite point of view, in a third approach we start with a particular integral transform, and then ask if there is a sequence of points such that its kernel yields a basis of some kind by the procedure described. Here the interst centres on the integral transform itself. A special case will be given which illustrates a connection with the theory of Watson transforms. Here there are many interesting, but at the moment open, problems.

Other specific sampling theorems will be mentioned in context, as time permits; they will illustrate various aspects of the theory.

Date received: July 7, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-61.