$L_p$ Stability of the Truncated Hierarchical B-Spline Basis
Received:May 25, 2017  Revised:September 01, 2017
Key Word: hierarchical spline space   truncated hierarchical B-spline basis   stable basis   condition number of basis   partition of unity
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11290143; 11471066), Fundamental Research of Civil Aircraft (Grant No.MJ-F-2012-04) and the Fundamental Research Funds for the Central Universities (Grant No.DUT15LK44).
 Author Name Affiliation Jianping ZHOU Department of Mathematics, Dalian University of Technology, Liaoning 116024, P. R. China Renhong WANG Department of Mathematics, Dalian University of Technology, Liaoning 116024, P. R. China Chongjun LI Department of Mathematics, Dalian University of Technology, Liaoning 116024, P. R. China
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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.