A sampling theory for infinite weighted graphs

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.