Harmonic Analysis and Applications

June 4-8, 2018


"A Guided Tour of Decomposition Spaces"

Voigtlaender, Felix

We 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.

« back