  Finding efficient representations is one of the most challenging and heavily
sought problems in mathematics. Representation using shearlets recently
receives a lot of attention due to their desirable properties in both theory
and applications. Using the framework of frequency-based affine systems, in
this paper we introduce and systematically study affine shear tight frames
which include all known shearlet tight frames as special cases. Our results in
this paper will resolve several key questions on shearlets. We provide a
complete characterization for an affine shear tight frame and then use it to
obtain smooth affine shear tight frames with all their generators in the
Schwarz class. Though multiresolution analysis (MRA) is the foundation and key
feature of wavelet analysis for fast numerical implementation of a wavelet
transform, all the known shearlets so far do not possess any MRA structure and
filter banks. In order to study affine shear tight frames with MRA structure,
we introduce the notion of a sequence of affine shear tight frames and then we
provide a complete characterization for it. Based on our characterizations, we
present two different approaches, i.e., non-stationary and quasi-stationary,
for the construction of sequences of affine shear tight frames with MRA
structure such that all their generators are smooth (in the Schwarz class) and
they have underlying filter banks. Consequently, their associated transforms
can be efficiently implemented using filter banks similarly as a fast wavelet
transform does.
