TY  - JOUR
ID  - Jorgensen2011
LB  - Jorgensen2011
AU  - Jorgensen, Palle E. T.
TI  - A sampling theory for infinite weighted graphs
ST  - A sampling theory for infinite weighted graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 2
SP  - 209
EP  - 236
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.2.209
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.2.209
AB  - 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.
KW  - weighted graph
KW  - Hilbert space
KW  - Laplace operator
KW  - sampling
KW  - Shannon
KW  - white noise
KW  - Wiener transform
KW  - interpolation
SN  - 1232-9274
ER  -