  Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2
discontinuity curve, have by now become a standard model for measuring sparse
(non-linear) approximation properties of directional representation systems. It
was already shown that curvelets, contourlets, as well as shearlets do exhibit
(almost) optimally sparse approximation within this model. However, all those
results are only applicable to band-limited generators, whereas, in particular,
spatially compactly supported generators are of uttermost importance for
applications.
  In this paper, we now present the first complete proof of (almost) optimally
sparse approximations of cartoon-like images by using a particular class of
directional representation systems, which indeed consists of compactly
supported elements. This class will be chosen as a subset of shearlet frames --
not necessarily required to be tight -- with shearlet generators having compact
support and satisfying some weak moment conditions.
