Anisotropic multiscale systems on bounded domains

Abstract

We establish a construction of hybrid multiscale systems on a bounded domain ${\Omega } \subset \mathbb {R}^{2}$ consisting of shearlets and boundary-adapted wavelets, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct hybrid shearlet-wavelet systems that form frames for the Sobolev spaces $H^{s}({\Omega }),~s\in \mathbb {N} \cup \{0\}$ with controllable frame bounds and admit optimally sparse approximations for functions which are smooth apart from a curve-like discontinuity. Per construction, these systems allow to incorporate boundary conditions.