21. Februar 2018, 14.30 pm

Seminar Room, Wohllebengasse 12-14, ground floor

Describing sparsity properties of frames using function spaces - Felix Voigtländer

Katholische Universität Eichstätt-Ingolstadt, Lehrstuhl Wiss. Rechnen

Abstract:

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.