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 that better reflects the graph's symmetries. 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. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley 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）技术设计中的核心要素。当邻接矩阵具有多维特征空间时，基于一个能更好反映图对称性的特定特征基来构建GSP方法，是理想的选择。本文针对加权凯莱图邻接矩阵的谱分解，提供了一个明确且详尽的表示论阐述。我们的方法适用于所有加权凯莱图，无论其是否为拟阿贝尔群，并提供了从底层群表示的系数函数导出的特征值与特征向量的详细描述。接下来，我们将注意力转向在凯莱图上构建框架。框架是过完备的生成集，能确保信号重构系统稳定且具有潜在的冗余性。我们利用所提出的特征基来构建适用于凯莱图信号处理开发的框架。这些框架包括Frobenius--Schur框架和Cayley框架，我们对其进行了刻画，并提供了实际构建的步骤。