$L_p$ Stability of the Truncated Hierarchical B-Spline Basis

DOI：10.3770/j.issn:2095-2651.2017.06.006

 作者 单位 周健萍 大连理工大学数学科学学院, 辽宁 大连 116024 王仁宏 大连理工大学数学科学学院, 辽宁 大连 116024 李崇君 大连理工大学数学科学学院, 辽宁 大连 116024

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.

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.