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.

中文翻译：

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有界域上的各向异性多尺度系统

我们在由剪切波和边界自适应小波组成的有界域\（{\ Omega} \ subset \ mathbb {R} ^ {2} \）上建立了混合多尺度系统的构造，它们满足了几个有利于影像学应用的特性以及偏微分方程的数值分析。更确切地说，我们构建杂种剪切波小波系统，对于Sobolev空间形式帧\（H ^ {S}（{\欧米茄}），382 4 \在\ mathbb {N} \杯\ {0 \} \）与可控制的帧边界，并为除曲线状不连续性之外的平滑函数提供最佳的稀疏近似。根据构造，这些系统允许合并边界条件。