Harmonic and Applied Analysis by Stephan Dahlke Filippo Mari Philipp Grohs & Demetrio Labate

Author:Stephan Dahlke, Filippo Mari, Philipp Grohs & Demetrio Labate
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.59)

Banach Frames: The set is a Banach frame for which means that i) if and only if ;

ii)

iii) there exists a bounded, linear reconstruction operator from to such that

Remark 3.39.

By choosing, e.g., the U-dense set according to Lemma 3.37 and the analyzing shearlet according to Theorem 3.36, we observe that Theorem 3.38 holds for a huge class of analyzing vectors.

3.5 Structure of Shearlet Coorbit Spaces

In the last section, we have established new smoothness spaces, the shearlet coorbit spaces, and we have shown how associated atomic decompositions and Banach frames can be constructed. However, so far these spaces only exist as abstract spaces of distributions for which the extended voice transform has some specific decay. Then, of course, the next question is how these spaces really ‘look like,’ i.e., to understand the structure of shearlet coorbit spaces. In particular, in this section, we want to answer the following questions: What kinds of functions or distributions are really contained in these spaces? Are there nice dense subsets? And what are the relations to classical smoothness spaces such as Besov spaces? The shearlet coorbit spaces are definitely not Besov spaces since the Besov spaces are related to a different group, the affine group. Nevertheless, the shearlet group and the affine group share some common characteristics, so that it is expected that there are some relations between the different associated smoothness spaces. Indeed, in Subsection 3.5.2 we show first embedding theorems of shearlet coorbit spaces into (homogeneous) Besov spaces. Moreover, in the last Subsection 3.5.3 we answer the question how traces of shearlet coorbit spaces can be described. It turns out that the trace spaces are either Besov spaces or lower-dimensional shearlet coorbit spaces.

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