I just saw an interesting talk about shearlets on PyData Berlin.
One point when defining shearlets is the use of so called cartoon like functions, which are defined as
Let $f ∈ L^2(\mathbb{R}^2)$ with its support contained in the closed unit square such that $f$ can be written as $$f = f_0 + 1_Bf_1,$$ where $B ⊂ [0, 1]^2$ is a set with a closed $C^2$ boundary curve and $\|f_0\|_{C_2}, \|f_1\|_{C_2} ≤ 1$.
So, I understand why we want this function, we need this assumptions later, compare this talk pages. But now I questioned myself: why are these functions named like this?
Maybe an explanation of what exactly these norms are would help, I don't get that index.
Thx in advance!
The main point is that $f_0$ and $f_1$ are smooth (do continuous, in fact twice differentiable), so that $f $ itself is smooth away from the singularity curve $\partial B $.
If you look at a typical cartoon (like this one), then you see that for cartoons essentially the same holds: You have certain boundaries (e.g. the boundary of a face), but away from these boundaries, the colors change in in a continuous (or even smooth way).
Why exactly one uses $C^2$ regularity instead of e.g. $C^1$ (or $C^\infty $) regularity, I can not really tell you; one reason is surely that this regularity makes the proofs work (i.e., only with this regularity, shearlets are optimal for approximating this type of functions).
EDIT: To explain these norms: We simply have $\| f \|_{C^2} = \sum_{|\alpha|\leq 2} \|\partial^\alpha f\|_{L^\infty} $, where the sum is over all multiindices of order $\leq 2$. But again, the assumption $\|f \|_{C^2}\leq 1$ is essentially arbitrary. Replacing the $1$ by $1000000$ would lead to the same results, but one has to choose something. The idea is to fix some quantitative form of regularity which one can then use for the proofs later on. For the Intuition, it is only important that $f_0,f_1$ are somewhat smooth.