Average sampling and average splines on combinatorial graphs
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In the setting of a weighted combinatorial finite or infinite countable graph G we introduce functional Paley-Wiener spaces PW_ω(L),ω>0, defined in terms of the spectral resolution of the combinatorial Laplace operator L in the space L_2(G). It is shown that functions in certain PW_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∈ PW_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.
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