  Graphons are limit objects of sequences of graphs and are used to analyze the
behavior of large graphs. Recently, graphon signal processing has been
developed to study signal processing on large graphs. A major limitation of
this approach is that any sparse sequence of graphs inevitably converges to the
zero graphon, rendering the resulting signal processing theory trivial and
inadequate for sparse graph sequences. To overcome this limitation, we propose
a new signal processing framework that leverages the concept of generalized
graphons and introduces the stretched cut distance as a measure to compare
these graphons. Our framework focuses on the sampling of graph sequences from
generalized graphons and explores the convergence properties of associated
operators, spectra, and signals. Our signal processing framework provides a
comprehensive approach to analyzing and processing signals on graph sequences,
even if they are sparse. Finally, we discuss the practical implications of our
theory for real-world large networks through numerical experiments.
