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Finding efficient representations is one of the most challenging and heavilysought problems in mathematics. Representation using shearlets recentlyreceives a lot of attention due to their desirable properties in both theoryand applications. Using the framework of frequency-based affine systems, inthis paper we introduce and systematically study affine shear tight frameswhich include all known shearlet tight frames as special cases. Our results inthis paper will resolve several key questions on shearlets. We provide acomplete characterization for an affine shear tight frame and then use it toobtain smooth affine shear tight frames with all their generators in theSchwarz class. Though multiresolution analysis (MRA) is the foundation and keyfeature of wavelet analysis for fast numerical implementation of a wavelettransform, all the known shearlets so far do not possess any MRA structure andfilter 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 weprovide a complete characterization for it. Based on our characterizations, wepresent two different approaches, i.e., non-stationary and quasi-stationary,for the construction of sequences of affine shear tight frames with MRAstructure such that all their generators are smooth (in the Schwarz class) andthey have underlying filter banks. Consequently, their associated transformscan be efficiently implemented using filter banks similarly as a fast wavelettransform does.

Smooth affine shear tight frames with MRA structure