  Topological signal processing (TSP) over simplicial complexes typically
assumes observations associated with the simplicial complexes are real scalars.
In this paper, we develop TSP theories for the case where observations belong
to general abelian groups, including function spaces that are commonly used to
represent time-varying signals. Our approach generalizes the Hodge
decomposition and allows for signal processing tasks to be performed on these
more complex observations. We propose a unified and flexible framework for TSP
that expands its applicability to a wider range of signal processing
applications. Numerical results demonstrate the effectiveness of this approach
and provide a foundation for future research in this area.
