We present a systematic approach towards understanding the sparsity properties of different frame constructions like Gabor systems, wavelets, shearlets, and curvelets. We use the following terminology: Analysis sparsity means that the frame coefficients are sparse (in an \ell^p sense), while synthesis sparsity means that the function can be written as a linear combination of the frame elements using sparse coefficients. While these two notions are completely distinct for general frames, we show that if the frame in question is sufficiently nice, then both forms of sparsity of a function are equivalent to membership of the function in a certain decomposition space.
These decomposition spaces are a common generalization of Besov spaces and modulation spaces. While Besov spaces can be defined using a dyadic partition of unity on the Fourier domain, modulation spaces employ a uniform partition of unity, and general decomposition spaces use an (almost) arbitrary partition of unity on the Fourier domain. To each decomposition space, there is an associated frame construction: Given a generator, the resulting frame consists of certain translated, modulated and dilated versions of the generator. These are chosen so that the frequency concentration of the frame is similar to the frequency partition of the decomposition space. For instance, Besov spaces yield wavelet systems, while modulation spaces yield Gabor systems. We give conditions on the (possibly compactly supported!) generator of the frame which ensure that analysis sparsity and synthesis sparsity of a function are both equivalent to membership of the function in the decomposition space.