Sampling by averages and average splines on Dirichlet spaces and on combinatorial graphs

Abstract: In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_{\omega},>>\omega>0,$ of Paley-Wiener functions of bandwidth $\omega$. It is shown that every function in $PW_{\omega},>>\omega>0,$ is uniquely determined by its average values over a family of balls $B(x_{j}, \rho),>x_{j}\in {\bf X},$ which form an admissible cover of ${\bf X}$ and whose radii are comparable to $\omega^{-1/2}$. The entire development heavily depends on some local and global Poincar\'e-type inequalities. In the second part of the paper we realize the same idea in the setting of a weighted combinatorial finite or infinite countable graph $G$. We have to treat the case of graphs separately since the Poincar\'e inequalities we are using on them are somewhat different from the Poincar\'e inequalities in the first part.