文献类型：期刊文献

英文题名：Approximation order equivalence properties of manifold-valued data subdivision schemes

作者：Xie, Gang[1];Yu, Thomas P-Y.[2]

机构：[1]E China Univ Sci & Technol, Dept Math, Shanghai 200237, Peoples R China;[2]Drexel Univ, Dept Math, Korman Ctr 206, Philadelphia, PA 19104 USA

年份：2012

卷号：32

期号：2

起止页码：687

外文期刊名：IMA JOURNAL OF NUMERICAL ANALYSIS

收录：;EI(收录号：20214911261391);WOS:【SCI-EXPANDED(收录号:WOS:000302500900013)】；

基金：The first named author was sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education, P. R. China. The second named author was supported by the National Science Foundation (grant DMS 0542237).

语种：英文

外文关键词：approximation order; subdivision scheme; nonlinear subdivision scheme; interpolation; quasiinterpolation; manifold

摘要：There has been emerging interest in developing an approximation theory for manifold-valued functions. In this paper we address the following fundamental problem: let M be a manifold with a metric d. For each smoothness factor r > 0 and approximation order R > 0, is there an approximation operator A(h) = A(h;r,R) that maps samples of any f: R -> M on a grid of size h to an approximant f(h) = A(h)(f |h(Z)): R -> M with the properties that (a) sup(x)d(f(h)(x), f (x)) = O(h(R)) whenever f is a bounded C-R function and (b) f(h) is C-r smooth? The case of M = R is of course well studied. In the recent paper (Xie, G. & Yu, T.P.-Y. (2008) Smoothness equivalence properties of general manifold-valued data subdivision schemes. Multiscale Model. Simul., 7, 1073-1100) the authors show that subdivision methods can be used to create arbitrarily smooth interpolants for M-valued data, addressing (b) above. In this paper we further show that interpolatory subdivision schemes can be used to solve (a) above. So, altogether, we establish the fact that if a linear interpolatory subdivision scheme possesses a smoothness order r and an approximation order R, then there is a construction of a nonlinear interpolatory subdivision scheme for M-valued data based on this linear scheme with the same smoothness and approximation orders. In other words subdivision schemes furnish a constructive approximation method for solving the open problem posted above. We discuss the construction of quasiinterpolants of manifold-valued data based on general (not necessarily interpolatory) subdivision schemes.