A few questions concerning shearlets

Question

I am currently reading Compactly supported shearlets are optimally sparse by Kutyniok and Lim and have a few small questions, which I have to give a talk about. At our university, this seminar is custom so one can take a dive into scientific literature before writing a bachelors thesis.

1. On page 2 they say that "However, customarily, at that time [referring to 10 years ago, the paper was published in 2011] an image was viewed as an element of a compact subset of $L^p$ characterized by a given Besov regularity with the Kolmogorov entropy of such sets identifying lower bounds for the distortion rates of encoding–decoding pairs in this model."

I know what distortion and encoding–decoding pairs are but had a hard time finding out how Kolmogorov entropy identifies lower bounds. Could someone please give a reference or short explanation on this method? As I understand it, Besov spaces are generalisations of Sobolev spaces, the latter of which I know about.

2. I am not a native speaker and wonder what it precisely means to "resolve an edge". It is used in all the literature I could find but explained nowhere. In the paper, it is used as follows: "About 10 years ago, mathematicians started to design models of images incorporating those findings aiming at designing representation systems which – in such a model – are capable of resolving edges in an optimally sparse way." (p. 1 - 2).

3. Also, separable shearlet frames are mentioned but never explained. In this short note on separable frames by Themba Dube I found the following definition: A frame $L$ is called separable if there is a countable set $S \subset L - \{1\}$ called a separator such that each nonzero element of $L$ joins some member of $S$ at the top. This is the only definition I found so far and it uses a lot of terminology I am unfamiliar with so I am not sure if this is the definition I am looking for. I know when spaces are separable but fail to see how separable frames are somehow similar.

4. Also, I am unfamiliar with the following notation: $$L^2(\{ f \in L^2(\mathbb{R}^2): \text{supp}(\hat{f}) \subset \mathcal{C}_1 \cup \mathcal{C}_3\})$$ appearing on page 1569. I only know $L^2(\Omega)$, where $\Omega$ is some subset of $\mathbb{R}^n$ but how is $L^2$ of a function class defined?