Exact positive cubature formulas via generalized sampling on combinatorial graphs

TL;DR

This paper develops exact positive cubature formulas for bandlimited signals on graphs using generalized sampling over graph partitions, enabling precise integration based on weighted averages.

Contribution

It introduces a novel approach to construct exact positive cubature formulas on graphs through generalized sampling on graph partitions.

Findings

Derived conditions for exact quadrature formulas on graphs.

Established positive weights for sampling-based integration.

Validated formulas for bandlimited functions on graphs.

Abstract

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.