Opuscula Math. 31, no. 2 (2011), 209-236
A sampling theory for infinite weighted graphs
Abstract. We prove two sampling theorems for infinite (countable discrete) weighted graphs \(G\); one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum \(X\) containing \(G\), and there are Hilbert spaces of functions on \(X\) that allow interpolation by sampling values of the functions restricted only on the vertices in \(G\). We sample functions on \(X\) from their discrete values picked in the vertex-subset \(G\). We prove two theorems that allow for such realistic ambient spaces \(X\) for a fixed graph \(G\), and for interpolation kernels in function Hilbert spaces on \(X\), sampling only from points in the subset of vertices in \(G\). A continuum is often not apparent at the outset from the given graph \(G\). We will solve this problem with the use of ideas from stochastic integration.
Keywords: weighted graph, Hilbert space, Laplace operator, sampling, Shannon, white noise, Wiener transform, interpolation.
Mathematics Subject Classification: 05C22, 68R01, 81T05, 42A99, 47L60, 94A20, 47B35.