Basic Sampling Theory

This chapter intends to serve as an introduction to sampling theory which deals with the reconstruction of functions through their values on an appropriate sequence of points by means of sampling expansions involving these values. Reproducing kernel Hilbert spaces are suitable spaces for sampling purposes since evaluation functional are continuous and thus, the recovery depends on the frame properties of the reproducing kernel at the sampling points. After an introduction to the basic theory of reproducing kernel Hilbert spaces, we pay attention to the paradigmatic example of Paley-Wiener spaces: their main properties and some generalizations related to sampling aspects are included.