Anisotropic Multiscale Systems on Bounded Domains

In this paper we provide a construction of multiscale systems on a bounded domain Ω ⊂ R coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and numerical analysis of partial differential equations. More precisely, we construct a boundary shearlet system that forms a frame for L(Ω) with controllable frame bounds and admits optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuities. Indeed, the constructed system allows for boundary conditions, and characterizes Sobolev spaces over Ω by its analysis coefficients. Finally, we demonstrate numerically that this system also constitutes a Gelfand frame for (H(Ω), L(Ω), H−s(Ω)) for s ∈ N.