Function Spaces for Sampling Expansions

Abstract

In this chapter, we consider a variety of Hilbert and Banach spaces that admit sampling expansions \(f(t) = \sum\nolimits_{n=1}^{\infty }f({t}_{n}){S}_{n}(t)\), where {S n } n=1 ∞ is a family of functions that depend on the sampling points {t n } n=1 ∞ but not on the function f. Those function spaces, that arise in connection with sampling expansions, include reproducing kernel spaces, Sobolev spaces, Wiener amalgam space, shift-invariant spaces, translation-invariant spaces, and spaces modeling signals with finite rate of innovation. Representative sampling theorems are presented for signals in each of these spaces. The chapter also includes recent results on nonlinear sampling of signals with finite rate of innovation, convolution sampling on Banach spaces, and certain foundational issues in sampling expansions.