The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis (the `preferred basis'). In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a (weighted) Cayley graph, which results in a preferred basis. Our method applies to all weighted (not necessarily quasi-Abelian) Cayley graphs, and provides descriptions of eigenvalues and eigenvectors based on the coefficient functions of the representations of the underlying group. Next, we use such bases to build frames that are suitable for developing signal processing on such graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.

翻译：图邻接矩阵的谱分解是图信号处理（GSP）技术设计中的关键要素。当邻接矩阵具有多维特征空间时，基于特定特征基（即“优选基”）构建GSP方法具有重要价值。本文针对（加权）Cayley图的邻接矩阵，提供了明确且详细的表示论解释，由此得到一组优选基。该方法适用于所有加权（不必是拟阿贝尔的）Cayley图，并通过所在群表示系数的函数给出特征值与特征向量的描述。进一步，我们利用此类基构建适用于此类图上信号处理的框架。这些框架包括Frobenius-Schur框架与Cayley框架，我们对其性质进行刻画，并给出实际构造方案。