Opuscula Math. 31, no. 2 (2011), 209-236
http://dx.doi.org/10.7494/OpMath.2011.31.2.209

Opuscula Mathematica

# A sampling theory for infinite weighted graphs

Palle E. T. Jorgensen

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.

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