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

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions