Multivariate Shearlet Transform, Shearlet Coorbit Spaces and Their Structural Properties

Abstract

This chapter is devoted to the generalization of the continuous shearlet transform to higher dimensions as well as to the construction of associated smoothness spaces and to the analysis of their structural properties, respectively. To construct canonical scales of smoothness spaces, so-called shearlet coorbit spaces, and associated atomic decompositions and Banach frames we prove that the general coorbit space theory of Feichtinger and Grochenig is applicable for the proposed shearlet setting. For the two-dimensional case we show that for large classes of weights, variants of Sobolev embeddings exist. Furthermore, we prove that for natural subclasses of shearlet coorbit spaces which in a certain sense correspond to “cone-adapted shearlets” there exist embeddings into homogeneous Besov spaces. Moreover, the traces of the same subclasses onto the coordinate axis can again be identified with homogeneous Besov spaces. These results are based on the characterization of Besov spaces by atomic decompositions and rely on the fact that shearlets with compact support can serve as analyzing vectors for shearlet coorbit spaces. Finally, we demonstrate that the proposed multivariate shearlet transform can be used to characterize certain singularities.