  Graph signal processing is an emerging field which aims to model processes
that exist on the nodes of a network and are explained through diffusion over
this structure. Graph signal processing works have heretofore assumed knowledge
of the graph shift operator. Our approach is to investigate the question of
graph filtering on a graph about which we only know a model. To do this we
leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make
three key contributions to the emerging field of graph signal processing. We
show first that filters defined over the scaled adjacency matrix of a random
graph drawn from a graphon converge to filters defined over the Fredholm
integral operator with the graphon as its kernel. Second, leveraging classical
findings from the theory of the numerical solution of Fredholm integral
equations, we define the Fourier-Galerkin shift operator. Lastly, using the
Fourier-Galerkin shift operator, we derive a graph filter design algorithm
which only depends on the graphon, and thus depends only on the probabilistic
structure of the graph instead of the particular graph itself. The derived
graphon filtering algorithm is verified through simulations on a variety of
random graph models.
