  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.
