  We propose a framework for generalized sampling of graph signals that
parallels sampling in shift-invariant (SI) subspaces. This framework allows for
arbitrary input signals, which are not constrained to be bandlimited.
Furthermore, the sampling and reconstruction filters may be different. We
present design methods of the correction filter that compensate for these
differences and lead to closed form expressions in the graph frequency domain.
In this study, we consider two priors on graph signals: The first is a subspace
prior, where the signal is assumed to lie in a periodic graph spectrum (PGS)
subspace. The PGS subspace is proposed as a counterpart of the SI subspace used
in standard sampling theory. The second is a smoothness prior that imposes a
smoothness requirement on the graph signal. We suggest the use of recovery
techniques for when the recovery filter can be optimized and under a setting in
which a predefined filter must be used. Sampling is performed in the graph
frequency domain, which is a counterpart of "sampling by modulation" used in SI
subspaces. We compare our approach with existing sampling techniques on graph
signal processing. The effectiveness of the proposed generalized sampling
approach is validated numerically through several experiments.
