Average sampling and average splines on combinatorial graphs

In the setting of a weighted combinatorial finite or infinite countable graph G we introduce functional Paley-Wiener spaces P W_ω(L), ω > 0, defined in terms of the spectral resolution of the combinatorial Laplace operator L in the space L₂(G). It is shown that functions in certain P W_ω(L), ω > 0, are uniquely defined by their averages over some families of "small" subgraphs which form a cover of G. Reconstruction methods for reconstruction of an f ∈ P W_ω(L) from appropriate set of its averages are introduced. One method is using language of Hilbert frames. Another one is using average variational interpolating splines which are constructed in the setting of combinatorial graphs.