Exact positive cubature formulas via generalized sampling on combinatorial graphs

We consider a disjoint cover (partition) of an undirected weighted finite graph $G$ by $|J|$ connected subgraphs (clusters) $\{S_{j}\}_{j\in J}$ and select a function $\zeta_{j}\geq 0$ on each of the clusters. For a given signal $f$ on $G$ the set of its weighted average values samples is defined via inner products $\{\langle \zeta_{j}, f\rangle\}_{j\in J}$. The goal of the paper is to establish exact quadrature formulas with positive weights which are exploring these samples generated by bandlimited functions.