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

In the framework of a strictly local regular Dirichlet space $\mathbf{X}$ we introduce the subspaces $P{W}_{\omega },\omega >0,$ of Paley-Wiener functions of bandwidth $\omega$. It is shown that every function in $P{W}_{\omega },\omega >0,$ is uniquely determined by its average values over a family of balls $B(x_j,\rho ),x_j\in \mathbf{X},$ which form an admissible cover of $\mathbf{X}$ and whose radii are comparable to ${\omega }^{-1\mathrm{/}2}$. 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.