  Anisotropic decompositions using representation systems such as curvelets,
contourlet, or shearlets have recently attracted significantly increased
attention due to the fact that they were shown to provide optimally sparse
approximations of functions exhibiting singularities on lower dimensional
embedded manifolds. The literature now contains various direct proofs of this
fact and of related sparse approximation results. However, it seems quite
cumbersome to prove such a canon of results for each system separately, while
many of the systems exhibit certain similarities. In this paper, with the
introduction of the concept of sparsity equivalence, we aim to provide a
framework which allows categorization of the ability for sparse approximations
of representation systems. This framework, in particular, enables transferring
results on sparse approximations from one system to another. We demonstrate
this concept for the example of curvelets and shearlets, and discuss how this
viewpoint immediately leads to novel results for both systems.
