Exact positive quadrature formulas on combinatorial graphs via local average sampling

In this paper, we consider a disjoint cover (partition) of an undirected weighted finite graph [Formula: see text] by [Formula: see text] connected subgraphs (clusters) [Formula: see text] and select a function [Formula: see text] on each of the clusters. For a given signal [Formula: see text] on [Formula: see text], its weighted average samples are defined via inner products [Formula: see text]. The goal of this paper is to describe subspaces of bandlimited functions for which there exist quadrature formulas with positive coefficients based on weighted average samples.