Fourier transform of vector-valued graph signals

Classical Graph Signal Processing (GSP) provides a robust framework for analyzing signals on irregular domains, utilizing the graph Fourier transform as a cornerstone for spectral analysis and filtering. However, as data structures grow in complexity, there is an increasing need to handle multi-dimensional information. In this paper, we propose a generalization of the GSP framework by introducing vector-valued graph signals which take values in arbitrary Banach spaces. We define and investigate the fundamental operators of vertex-frequency analysis within this broader setting, including the Fourier transform, convolution, and translation operators. A key contribution of this work is the derivation of operator norm estimates and the establishment of graph-theoretic versions of classical uncertainty principles. We demonstrate how these results depend on the choice of the orthonormal basis and on the underlying $L^p$ norms. By modeling multiple scalar signals as a single vector-valued entity, this framework facilitates the study of inter-signal correlations, providing a flexible and mathematically grounded environment for analyzing multivariate time-series and time-varying signals on complex networks.