Analysis of shearlet coorbit spaces
- Analyse von Shearlet-Coorbit-Räumen
Koch, René; Führ, Hartmut (Thesis advisor); Dahlke, Stephan (Thesis advisor); Rauhut, Holger (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2018
Coorbit theory provides a framework for the study of approximation theoretic properties of certain elementary building blocks and to analyze properties of functions by considering the decay behavior of their associated wavelet transform. This transform is based on a representation of a group and properties of the group are decisive for the structure of the associated coorbit space, which is defined by imposing decay conditions on the wavelet transform. A recently established result that facilitates the study of coorbit spaces is given by their identification with decomposition spaces via the Fourier transform. This identification of coorbit spaces with certain decomposition spaces relies on a covering induced by the associated group. Since the classical wavelet transform lacks the ability of analyzing directional information reliably due to its isotropic nature, a different, directionally sensitive transform based on shearlet groups receives currently attention. This shearlet transform is able to analyze anisotropic information of signals. In this thesis, we apply and extend recently established results in connection with the identification of coorbit spaces with decomposition spaces and pay particular attention to the spaces based on shearlet groups in three and higher dimension. First, we construct explicitly induced coverings by these groups in three dimensions, which is a prerequisite for the identification with decomposition spaces. By studying the properties of these coverings and their relation to each other, we make the statement that distinct groups lead to distinct spaces precise. Building on that, we apply a general result about the embedding of decomposition spaces into Sobolev spaces to characterize the embedding of shearlet coorbit spaces in three dimensions into Sobolev spaces. It is desirable, particularly for higher dimensions, to establish methods to compare the decomposition spaces associated to different groups without having to compute coverings explicitly. To this end, we encode the geometric properties of different coverings via an induced metric. This metric defines a coarse structure on the orbit of the group and the relation of different coverings can be decided by examining the multiplicative structure of the groups.