We consider a disjoint cover (partition) of an undirected weighted finite or infinite graph G by J connected subgraphs (clusters) \(\{S_{j}\}_{j\in J}\) and select functions \(\psi _{j}\) 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 \(\{\langle f, \psi _{j}\rangle \}_{j\in J}\). The main results of the paper are based on Poincare-type inequalities that we introduce and prove. These inequalities provide an estimate of the norm of the signal f on the entire graph G from sets of samples of f and its local gradient on each of the subgraphs. This allows us to establish discrete Plancherel-Polya-type inequalities (or Marcinkiewicz-Zigmund-type or frame inequalities) for signals whose gradients satisfy a Bernstein-type inequality. These results enable the development of a sampling theory for signals on undirected weighted finite or infinite graphs. For reconstruction of the signals from their samples an interpolation theory by weighted average variational splines is developed. Here by a weighted average variational spline we understand a minimizer of a discrete Sobolev norm which takes on the prescribed weighted average values on a set of clusters (in particular, just values on a subset of vertices). Although our approach is applicable to general graphs it’s especially well suited for finite and infinite graphs with multiple clusters. Such graphs are known as community graphs and they find many important applications in materials science, engineering, computer science, economics, biology, and social studies.

我们考虑由J个相连的子图（簇）\（\ {S_ {j} \} _ {j \ in J} \）构成的无向加权有限或无限图G的不相交覆盖（分区），并选择函数\（\ psi _ {j} \）在每个群集上。对于G上的给定信号f，通过内部乘积\（\ {\ langle f，\ psi _ {j} \ rangle \} _ {j \ in J} \）定义其加权平均值样本集。本文的主要结果是基于我们引入并证明的Poincare型不等式。这些不等式提供信号的范数的估计˚F上整个图形ģ从套样本˚F以及每个子图上的局部梯度。这使我们能够为梯度满足伯恩斯坦型不等式的信号建立离散的Plancherel-Polya型不等式（或Marcinkiewicz-Zigmund型或框架不等式）。这些结果使无方向加权有限图或无限图上的信号采样理论的发展成为可能。为了从其样本中重建信号，开发了一种基于加权平均变异样条的插值理论。在这里，通过加权平均变化样条曲线，我们了解了离散Sobolev范数的极小值，它在一组聚类上采用规定的加权平均值（特别是仅在顶点子集上的值）。尽管我们的方法适用于一般图，但它特别适合于具有多个群集的有限图和无限图。