Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

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