On Polynomial Approximations of Spectral Windows in Vertex-Frequency Representations

Vertex-frequency analysis (VF) can be considered as a generalization of the classical time-frequency analysis. It provides tools and algorithms aiming to characterize the localized signal behavior in the joint vertex-frequency domain. Localized Graph Fourier Transform (LGFT) is an example of such a tool, with a role in the graphs signal processing which is equivalent to the role of the Short-time Fourier transform in traditional signal processing. Bearing in mind the rapidly increasing amounts of data and large dimensions of graphs related to practical applications, the calculation complexity of each tool for the spectral analysis of signals on graphs shall be continuously revisited. As they provide the possibility to calculate VF representations using only local neighborhoods of vertices, without the need for the eigendecomposition, polynomial approximations of spectral windows are commonly used in practice, mostly in the form of the Chebychev approximation. This paper revisits this choice, compares it with two other polynomial approximation approaches, and investigates their influence on the VF-based graph signal analysis and inversion.