Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

We study kernel functions, and associated reproducing kernel Hilbert spaces 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 article, 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.