Understanding X-let sparsity via decomposition spaces

We present the recent theory of structured Banach frame decompositions for decomposition spaces, which helps understand the sparsity properties of a wealth of different types of frames, in particular of Gabor, wavelet and shearlet frames. The theory starts with a covering Q of the frequency space ℝ
^d
, which gives rise to a family of decomposition spaces. Furthermore, the covering Q also prescribes a mapping ψ ı→ Ψ
_δ
which associates a certain generalized shift invariant system Ψ
_δ
to a given finite set of generators ψ: For the uniform covering, Ψ
_δ
is the Gabor system generated by ψ, while the usual dyadic covering yields wavelet systems. Our main result is a set of conditions concerning the generators which ensure that Ψ
_δ
forms a Banach frame and an atomic decomposition simultaneously for a whole range of decomposition spaces, for sufficiently small sampling density δ > 0. In particular, membership of a signal f in one of these decomposition spaces is simultaneously equivalent to analysis sparsity of f and to synthesis sparsity of f with respect to the family Ψ
_δ
, where the precise notion of sparsity is determined by the parameters of the decomposition space. In most cases, the conditions concerning the generators ψ can be satisfied for suitable compactly supported generators. Even though the presented theory is similar to coorbit theory in many aspects, it applies in some important cases where coorbit theory does not. In particular, the presented theory yields equivalence of analysis and synthesis sparsity for compactly supported cone-adapted shearlet frames, a novel result with applications concerning the nonlinear approximation of cartoonlike functions.