21. February 2018,
Seminar Room, Wohllebengasse 12-14, ground floor
We present a systematic approach towards understanding the sparsity properties
of different frame constructions like Gabor systems, wavelets, shearlets, and curvelets.
Here, analysis sparsity means that the frame coefficients are sparse
(in an l^p sense), while synthesis sparsity means that the function can be
written as a linear combination of the frame elements using sparse coefficients.
We show that 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 Besov spaces, one obtains 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.