  Graph signal processing deals with signals which are observed on an irregular
graph domain. While many approaches have been developed in classical graph
theory to cluster vertices and segment large graphs in a signal independent
way, signal localization based approaches to the analysis of data on graph
represent a new research direction which is also a key to big data analytics on
graphs. To this end, after an overview of the basic definitions in graphs and
graph signals, we present and discuss a localized form of the graph Fourier
transform. To establish an analogy with classical signal processing, spectral-
and vertex-domain definitions of the localization window are given next. The
spectral and vertex localization kernels are then related to the wavelet
transform, followed by a study of filtering and inversion of the localized
graph Fourier transform. For rigour, the analysis of energy representation and
frames in the localized graph Fourier transform is extended to the energy forms
of vertex-frequency distributions, which operate even without the need to apply
localization windows. Another link with classical signal processing is
established through the concept of local smoothness, which is subsequently
related to the particular paradigm of signal smoothness on graphs. This all
represents a comprehensive account of the relation of general vertex-frequency
analysis with classical time-frequency analysis, and important but missing link
for more advanced applications of graphs signal processing. The theory is
supported by illustrative and practically relevant examples.
