The Graph Fractional Fourier Transform in Hilbert Space

Graph signal processing (GSP) leverages the inherent signal structure within graphs to extract high-dimensional data without relying on translation invariance. It has emerged as a crucial tool across multiple fields, including learning and processing of various networks, data analysis, and image processing. In this paper, we introduce the graph fractional Fourier transform in Hilbert space (HGFRFT), which provides additional fractional analysis tools for generalized GSP by extending Hilbert space and vertex domain Fourier analysis to fractional order. First, we establish that the proposed HGFRFT extends traditional GSP, accommodates graphs on continuous domains, and facilitates joint time-vertex domain transform while adhering to critical properties such as additivity, commutativity, and invertibility. Second, to process generalized graph signals in the fractional domain, we explore the theory behind filtering and sampling of signals in the fractional domain. Finally, our simulations and numerical experiments substantiate the advantages and enhancements yielded by the HGFRFT.