  We present novel families of wavelets and associated filterbanks for the
analysis and representation of functions defined on circulant graphs. In this
work, we leverage the inherent vanishing moment property of the circulant graph
Laplacian operator, and by extension, the e-graph Laplacian, which is
established as a parameterization of the former with respect to the degree per
node, for the design of vertex-localized and critically-sampled higher-order
graph (e-)spline wavelet filterbanks, which can reproduce and annihilate
classes of (exponential) polynomial signals on circulant graphs. In addition,
we discuss similarities and analogies of the detected properties and resulting
constructions with splines and spline wavelets in the Euclidean domain.
Ultimately, we consider generalizations to arbitrary graphs in the form of
graph approximations, with focus on graph product decompositions. In
particular, we proceed to show how the use of graph products facilitates a
multi-dimensional extension of the proposed constructions and properties.
