  We propose an inexact method for the graph Fourier transform of a graph
signal, as defined by the signal decomposition over the Jordan subspaces of the
graph adjacency matrix. This method projects the signal over the generalized
eigenspaces of the adjacency matrix, which accelerates the transform
computation over large, sparse, and directed adjacency matrices. The trade-off
between execution time and fidelity to the original graph structure is
discussed. In addition, properties such as a generalized Parseval's identity
and total variation ordering of the generalized eigenspaces are discussed.
  The method is applied to 2010-2013 NYC taxi trip data to identify traffic
hotspots on the Manhattan grid. Our results show that identical highly
expressed geolocations can be identified with the inexact method and the method
based on eigenvector projections, while reducing computation time by a factor
of 26,000 and reducing energy dispersal among the spectral components
corresponding to the multiple zero eigenvalue.
