The Sampling Theorem for Functions with Limited Multi-Band Spectrum I

Abstract

In this paper functions $f$ belonging to $L^2(\mathbb{R})$ are considered which spectrum is contained in a ’multi-band’ set $E$, i.e. in a subset of the real axis which is the union of finite many intervals. For such functions a generalization of the Whittaker-Kotelnikov-Shannon sampling formula is given. The considered problem is also related to Riesz bases of exponentials in $L^2(E)$. In the first part of this work we consider sets $E$ consisting of regularly positioned intervals of the same length.