Democracy of shearlet bases with applications to approximation and interpolation

Shearlets on the cone provide Parseval frames for $L^2$. They also provide near-optimal approximation for the class $\mathcal{E}$ of cartoon-like images. Moreover, there are spaces associated to them other than $L^2$ and there exist embeddings between these and classical spaces. We prove approximation properties of the cone-adapted shearlets in a more general context, namely, when the target function belongs to a class or space different to $\mathcal{E}$ and when the error is not necessarily measured in the $L^2$-norm but in a much wider family of smoothness space of high anisotropy.