GRAPH LAPLACIANS AND DISCRETE REPRODUCING KERNEL HILBERT SPACES FROM RESTRICTIONS AND CAMERON-MARTIN

We study reproducing kernel functions, and associated reproducing kernel Hilbert spaces (RKHSs) 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 PDE-boundary value problems; typically using multiresolution-subdivision schemes, applied to the 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 PDE-boundary value problems, but now serving as a tool for obtaining solutions in the discrete world.