Democracy of shearlet bases with applications to approximation and interpolation

TL;DR

This paper studies shearlet systems on the cone, demonstrating their approximation capabilities for various function spaces beyond $L^2$, and explores their applications in approximation and interpolation.

Contribution

It extends the approximation properties of cone-adapted shearlets to broader function spaces and error measures, beyond the classical $L^2$ setting.

Findings

Shearlets form near-optimal approximations for cartoon-like images.

They provide embeddings between different function spaces.

Approximation properties are established in more general contexts.

Abstract

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.