Sampling Theory for Functions with Fractal Spectrum

Abstract

We investigate in greater detail a sampling formula given by the first author for functions whose spectrum lies in a Cantor set $K$ of a special type introduced by Jorgensen and Pedersen, where the sampling set is extremely thin, and the sampling function is quite different from the usual sinc function. We obtain new properties of the sampling function, and we give approximate descriptions of both local and global behavior of functions with spectrum in $K$. Some experimental results are described, and more can be found at http://mathlab.cit.cornell.edu/tillman.