{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:59Z","timestamp":1787329319890,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Subdivision is a powerful way of approximating a continuous object $f(x,y)$ by a sequence $( (S^lp_{i,j})_{i,j\\in \\mathbb{Z}})_{l\\in \\mathbb{N}}$ of discrete data on finer and finer grids. The rule S that maps an approximation on a coarse grid $S^lp$ to the approximation on the next finer grid $S^{l+1}p$ is called a subdivision scheme. If for a given scheme S every continuous object $f(x,y)$ constructed by S is of $C^k$ smoothness, then S is said to have smoothness order k. Subdivision schemes are well understood if they are linear. However, for various applications the data have values in a manifold which is not a vector space (for example, when our data are positions of a moving rigid body). Under these circumstances, subdivision schemes become nonlinear and much harder to analyze. One way of analyzing such schemes is to relate them to a given linear scheme and to establish a so-called proximity condition between the two schemes, which helps in proving that the two schemes share the same smoothness. The present paper uses this method to show the $C^1$-smoothness of a wide class of nonlinear multivariate schemes.<\/jats:p>","DOI":"10.1137\/060669759","type":"journal-article","created":{"date-parts":[[2008,5,4]],"date-time":"2008-05-04T04:04:04Z","timestamp":1209873844000},"page":"2169-2182","source":"Crossref","is-referenced-by-count":23,"title":["Smoothness Analysis of Subdivision Schemes on Regular Grids by Proximity"],"prefix":"10.1137","volume":"46","author":[{"given":"Philipp","family":"Grohs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,2]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. S. Cavaretta, C. A. Micchelli, and W. Dahmen,\n                      Stationary Subdivision\n                      , American Mathematical Society, Boston, MA, 1991.","DOI":"10.1090\/memo\/0453"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"D. L. Donoho and T. P. Yu,\n                      Deslauriers-Dubuc: Ten years after\n                      , in Spline Functions and the Theory of Wavelets, S. Dubuc and G. Deslauriers, eds., CRM Proc. Lecture Notes 18, American Mathematical Society, Providence, RI, 1999, pp. 355\u2013370.","DOI":"10.1090\/crmp\/018\/29"},{"key":"R3","unstructured":"D. L. Donoho,\n                      Interpolating Wavelet Transforms\n                      , Technical report, Stanford University, Stanford, CA, 1992; also available from http:\/\/www-stat.stanford.edu\/~donoho\/Reports\/1992\/interpol.pdf."},{"key":"R4","unstructured":"D. L. 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Grohs,\n                      Smoothness Analysis of Nonlinear Subdivision Schemes on Regular Grids\n                      , Ph.D. thesis, Vienna University of Technology, Wien, Austria, 2007."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1111\/1467-8659.1530409"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.2307\/1969989"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/050622729"},{"key":"R11","unstructured":"M. Spivak,\n                      A Comprehensive Introduction to Differential Geometry\n                      , Vol. I\u2013V, Publish or Perish, Boston, MA, 1979."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.06.003"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/060668353"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-006-0638-3"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/060652944"},{"key":"R16","unstructured":"T. P.Y. Yu,\n                      Cutting corners on the sphere\n                      , in Wavelets and Splines: Athens 2005, G. Chen and M.J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 496\u2013506."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0527047"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060669759","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:37Z","timestamp":1787325277000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060669759"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060669759"],"URL":"https:\/\/doi.org\/10.1137\/060669759","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}