On stability of sampling-reconstruction models

Abstract

Abstract. A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this paper we prove this result for a large class of sampling models. We define different classes of perturbations and quantify the robustness of a model with respect to them. We also use the theory of localized frames to study the frame algorithm for recovering the original signal from its samples.

A sampling theory for infinite weighted graphs

“…In more detail, starting with a fixed infinite graph G, it will be convenient for us to denote the set of vertices G (0) , and the edges G (1) . And we will study functions on both sets; more precisely, Hilbert spaces obtained by completion in certain quadratic forms; as well as the interconnections between spaces of functions on G (0) , and on G (1) .…”

“…In this case, the interconnections between spaces may be understood with the use of the two rules for electrical networks, Ohm's law, and Kirchhoff's sum-rules for current flow. In this case, therefore functions on G (0) represent voltages, for example voltage potentials; while functions on G (1) can be a configuration of Kirchhoff-current (measured in Amps).…”

“…We are looking at a positive function c (the notation c is for conductance, and c = 1/resistance). We define a weighted graph to be a pair (G, c) where c is a fixed positive function defined on the set of (unordered) edges G (1) .…”

“…There is a recent renewed interest in sampling techniques, motivated in part by quantization requirements in digital signal processing, and by contrast to our present approach, much of this is based on harmonic analysis tools, see e.g., [1,2,6,13,14], and other investigations on Markov processes [15, 18-20, 22, 23, 26].…”

“…We show that there are important function spaces amenable to precisely the kind of interpolation, discussed in the previous sections, and envisioned by Shannon in the context of Fourier analysis. Following notation from section 2, we will denote a weighted graph (G, c), but as noted, it consists of three items, two sets G (0) and G (1) , and a function…”