{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:23:51Z","timestamp":1787322231439,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized 3D cartoon-like images. This function class will have two smoothness parameters: one parameter $\\beta$ controlling classical smoothness and one parameter $\\alpha$ controlling anisotropic smoothness. The class then consists of piecewise $C^\\beta$-smooth functions with discontinuities on a piecewise $C^\\alpha$-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the 3D setting and construct frames for $L^2(\\mathbb{R}^3)$ with this particular shearlet structure. For the smoothness range $1&lt;\\alpha\\le \\beta\\le 2$ we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of nonlinear $N$-term approximations.<\/jats:p>","DOI":"10.1137\/110844726","type":"journal-article","created":{"date-parts":[[2012,8,16]],"date-time":"2012-08-16T14:24:34Z","timestamp":1345127074000},"page":"2962-3017","source":"Crossref","is-referenced-by-count":44,"title":["Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames"],"prefix":"10.1137","volume":"44","author":[{"given":"Gitta","family":"Kutyniok","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jakob","family":"Lemvig","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wang-Q","family":"Lim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,8,16]]},"reference":[{"key":"atypb1","unstructured":"R. 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