Orthogonal Sampling Formulas: A Unified Approach

This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values. In order to obtain such sampling expansions in a unified way, we propose an inductive procedure leading to various orthogonal formulas. This procedure, which we illustrate with a number of examples, closely parallels the theory of orthonormal bases in a Hilbert space. All intermediate steps will be described in detail, so that the presentation is self-contained. The required mathematical background is a basic knowledge of Hilbert space theory. Finally, despite the introductory level, some hints are given on more advanced problems in sampling theory, which we motivate through the examples.