Approximation order equivalence properties of manifold-valued data subdivision schemes
Approximation order equivalence properties of manifold-valued data subdivision schemes
复制标题
流形数据细分方案的近似阶等价性质
DOI：
10.1093/imanum/drq046
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发表时间：
2012
影响因子：
2.1
通讯作者：
T. Yu
中科院分区：
文献类型：
--
作者：
G. Xie;T. Yu
人们对开发多值函数的近似理论产生了浓厚的兴趣。在本文中，我们解决了以下基本问题：令 M 为具有度量 d 的流形。对于每个平滑因子 r > 0 和近似阶数 R > 0，是否存在一个近似算子 Ah = Ah;r,R 将大小为 h 的网格上的任何 f : R→ M 的样本映射到近似 fh = Ah(f |hZ) : R→ M，其属性为：a) supx d(fh(x), f(x)) = O(h)，只要 f 是有界 C 函数，b) fh 是 C 平滑？ M = R 的情况当然是经过充分研究的。在最近的论文 [14] 中，作者表明细分方法可用于为 M 值数据创建任意平滑插值，解决上述 b) 问题。在本文中，我们进一步证明插值细分方案可以用于解决上述a）问题。因此，总的来说，我们确定了这样一个事实：如果线性插值细分方案具有平滑度 r 和近似阶 R，则基于该线性方案构造的 M 值数据的非线性插值细分方案具有相同的平滑度和近似阶数。换句话说，细分方案提供了一种构造性近似方法来解决上面提出的开放问题。我们讨论基于一般（不一定是插值）细分方案的流形值数据的准插值的构造。致谢。这项研究的工作得到了国家科学基金会拨款 DMS 0542237 的部分支持。近似等价的实证观察（图 1）首次在第六届国际曲线和曲面会议（法国阿维尼翁，2006 年 6 月 29 日至 7 月 5 日）上提出，并在第十二届国际近似理论会议（德克萨斯州圣安东尼奥，3 月）的小型研讨会上进一步讨论。 2007 年 4-8 日）和 MAIA 2007 会议（挪威奥勒松，2007 年 8 月 22-26 日）。[5] 对此问题做出了初步努力。第二位作者感谢 Nira Dyn 邀请他参加“细分和可精炼性”研讨会（意大利锡耶纳庞蒂尼亚诺，2008 年 5 月 1 日至 4 日），在会上首次介绍了本文的主要结果。
There has been an 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 Ah = Ah;r,R that maps samples of any f : R→ M on a grid of size h to an approximant fh = Ah(f |hZ) : R→ M with the properties that a) supx d(fh(x), f(x)) = O(h) whenever f is a bounded C function, and b) fh is C smooth ? The case of M = R is of course well-studied. In the recent paper [14], 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 a nonlinear interpolatroy subdivision scheme for M -valued data constructed based on this linear scheme has 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 quasi-interpolants of manifold-valued data based on general (not necessarily interpolatory) subdivision schemes. Acknowledgments. The work of this research was partially supported by the National Science Foundation grant DMS 0542237. The empirical observation on approximation equivalence (Figure 1) was first presented in the Sixth International Conference on Curves and Surfaces (Avignon, France, June 29-July 5, 2006.) and further discussed in a mini-symposium in the 12th International Conference in Approximation Theory (San Antonio, Texas, March 4-8, 2007) and the MAIA 2007 conference (Alesund, Norway, August 22-26, 2007.) A preliminary effort on this problem was made in [5]. The second named author thanks Nira Dyn for inviting him to a workshop on “Subdivision and Refinability” (Pontignano, Siena, Italy, May 1-4, 2008), at which the main result of this paper was first presented.