A Generalization of the Convolution Theorem and its Connections to Non-Stationarity and the Graph Frequency Domain

In this paper, we present a novel convolution theorem which encompasses the well known convolution theorem in (graph) signal processing as well as the one related to time-varying filters. Specifically, we show how a node-wise convolution for signals supported on a graph can be expressed as another node-wise convolution in a frequency domain graph, different from the original graph. This is achieved through a parameterization of the filter coefficients following a basis expansion model. After showing how the presented theorem is consistent with the already existing body of literature, we discuss its implications in terms of non-stationarity. Finally, we propose a data-driven algorithm based on subspace fitting to learn the frequency domain graph, which is then corroborated by experimental results on synthetic and real data.