Abstract
In the framework of a strictly local regular Dirichlet space X we introduce the subspaces PW ω, ω > 0, of Paley–Wiener functions of bandwidth ω. It is shown that every function in PW ω, ω > 0, is uniquely determined by its average values over a family of balls B(x j, ρ), x j ∈X, which form an admissible cover of X and whose radii are comparable to ω−1∕2. The entire development heavily depends on some Poincaré-type inequalities. In the second part of the paper we realize the same idea in the setting of a weighted combinatorial finite or infinite graph G. We have to treat the case of graphs separately since the Poincaré inequalities we are using on them are somewhat different from the Poincaré inequalities in the first part.
Dedicated to the 80th Birthday of John Benedetto.
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Pesenson, I.Z. (2021). Sampling by Averages and Average Splines on Dirichlet Spaces and on Combinatorial Graphs. In: Hirn, M., Li, S., Okoudjou, K.A., Saliani, S., Yilmaz, Ö. (eds) Excursions in Harmonic Analysis, Volume 6. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-69637-5_14
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