We establish a construction of hybrid multiscale systems on a bounded domain O⊂R2 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 Hs(O), s∈N∪{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.
