| 周健萍,王仁宏,李崇君.截断分层B样条基的Lp稳定性[J].数学研究及应用,2017,37(6):697~709 |
| 截断分层B样条基的Lp稳定性 |
| $L_p$ Stability of the Truncated Hierarchical B-Spline Basis |
| 投稿时间：2017-05-25 修订日期：2017-09-01 |
| DOI：10.3770/j.issn:2095-2651.2017.06.006 |
| 中文关键词: 分层样条空间 截断分层B样条基 稳定基 基底的条件数 单位分解性 |
| 英文关键词:hierarchical spline space truncated hierarchical B-spline basis stable basis condition number of basis partition of unity |
| 基金项目:国家自然科学基金(Grant Nos.11290143; 11471066), 民用飞机专项项目(Grant No.MJ-F-2012-04),中央高校基本科研业务费专项资金资助(Grant No.DUT15LK44). |
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| 中文摘要: |
| 截断分层B样条基是自适应曲线曲面拟合的重要的工具,它兼顾了宏观形状的构造和局部细节的控制.这种基底的$L_\infty$模的强稳定性在文献[20]中被讨论,但这种基底的$L_p(1\le p\le\infty)$模的稳定性并不清晰.在本文中我们讨论截断分层B样条基的$L_p$稳定性,既然基底的$L_p$稳定性对于形状构造十分重要.我们证得截断分层样条基是弱$L_p$稳定基,即基底稳定性中的系数依赖于分层空间的层数. |
| 英文摘要: |
| The truncated hierarchical B-spline basis has been proposed for adaptive data fitting and has already drawn a lot of attention in theory and applications. However the stability with respect to the $L_p$-norm, $\ 1\leq p < \infty$, is not clear. In this paper, we consider the $L_p$ stability of the truncated hierarchical B-spline basis, since the $L_p$ stability is useful for curve and surface fitting, especially for least squares fitting. We prove that this basis is weakly $L_p$ stable. This means that the associated constants to be considered in the stability analysis are at most of polynomial growth in the number of the hierarchy depth. |
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