Sampling in Complex and Harmonic Analysis

Abstract

This is a survey article on uniqueness, sampling and interpolation problems in complex analysis. Most of these problems are motivated by applications of great practical importance in signal analysis and data transmission, but they also admit other mathematical formulations relating them to fundamental questions about existence of good bases in function spaces. This circle of ideas in complex analysis has experimented in recent years a notorious revitalization, mostly because of its connections with analogous problems in time-frequency and wavelet analysis, some of which will be discussed as well. 1. Sampling and Interpolation for Band Limited Functions In this paper we will be talking mostly about sampling, and the most well-known sampling result is the one dealing with band-limited functions, known as the Shannon-Whitaker theorem (or Kotelnikov theorem in Russia). A τ-band-limited function is one with finite energy, i.e. |f (t)| 2 dt τ. Since f (t) = 1 √ 2π +τ −τ ˆ f (ζ)e itζ dζ, one may think f as being an (infinite) linear combination of sine and cosine functions with frequencies |ζ| < τ. We denote by B 2 τ the space of τ-band limited functions; note that the Fourier transform is an isometry between B 2 τ and L 2 (−τ, τ). The KSW sampling theorem states that the general form of such a function is given by the so called cardinal series f (t) = +∞ k=−∞ a k sin(τ t − πk) τ t − πk with k |a k | 2 < +∞. Note that a k = f kπ τ and, in fact, k |a k | 2 = τ π +∞ −∞ |f (t)| 2 dt .