**"A Guided Tour of Decomposition Spaces"**
#### Voigtlaender, FelixWe present recent progress in the theory of decomposition spaces.
These results provide a systematic approach towards understanding the
sparsity properties of different frame constructions including, but
not limited to Gabor systems, wavelets, alpha-Gabor-wavelet frames,
shearlets, and curvelets.
Generally, the following questions concerning the approximation theoretic
properties of a frame $\Gamma=(\gamma_i)_{i\in I}$ are of interest:
(1) Which signals can be sparsely encoded using $\Gamma$, i.e., for which
functions $f$ are the analysis coefficients $(\langle f,\gamma_i\rangle)_{i\in I}$
in $\ell^p$?
(2) Which signals can be sparsely represented using $\Gamma$, i.e., for
which functions $f$ can we write $f=\sum_{i\in I}c_i \gamma_i$
with $(c_i)_{i\in I}\in\ell^p$?
(3) Do the two signal classes from the preceding questions coincide?
(4) If a signal $f$ can be sparsely encoded using $\Gamma$, does this
tell us something about the integrability and/or smoothness of $f$?
(5) If a signal can be sparsely encoded using one frame (e.g., a wavelet
frame), does this imply that it can also be sparsely encoded using
a very different frame (e.g., a Gabor frame)?
We will see that if $\Gamma$ is a generalized shift invariant system
(GSI system) of a certain form, then these questions can be answered
by recent results in the theory of decomposition spaces. These spaces
are a generalization of both Besov spaces and modulation spaces: Recall
that Besov spaces can be defined using a dyadic partition of the Fourier
domain, while modulation spaces use a uniform partition. General decomposition
spaces are defined in the same way, but using an (almost) arbitrary
covering of the frequency domain. If one chooses the frequency covering
to be compatible with the frequency concentration of the frame, then
the resulting decomposition space can be used to answer the preceding
questions.
Instead of presenting the very abstract theory in full generality,
we will focus on the special cases of shearlets and $\alpha$-modulation
spaces to illustrate the use of decomposition spaces. Even in these
settings, our recent findings provide a unified perspective on various
well-known results; moreover, they answer several open questions,
in particular concerning the equivalence of questions (1) and (2)
from above. |