  We study kernel functions, and associated reproducing kernel Hilbert spaces
$\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical
analysis builds discrete models (e.g., finite element) for the purpose of
finding approximate solutions to boundary value problems; using
multiresolution-subdivision schemes in continuous domains. In this paper, we
turn the tables: our object of study is realistic infinite discrete models in
their own right; and we then use an analysis of suitable continuous counterpart
problems, but now serving as a tool for obtaining solutions in the discrete
world.
