Lp-stability of the truncated hierarchical B-spline basis
Received:May 25, 2017  Revised:August 18, 2017
Key Word: Hierarchical spline space   Truncated hierarchical B-spline basis   Stable basis   Condition number of basis   Partition of unity.  
Fund ProjectL:The National Natural Science Foundation of China (Nos. 11290143, No. 11471066), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04), and the Fundamental Research Funds for the Central Universities (No. DUT13LK07).
Author NameAffiliationE-mail
Jian-Ping Zhou School of Mathematical Sciences, Dalian University of Technology. jp.zhou@mail.dlut.edu.cn 
Ren-Hong Wang School of Mathematical Sciences, Dalian University of Technology. renhong@dlut.edu.cn 
Chong-Jun Li School of Mathematical Sciences, Dalian University of Technology. chongjun@dlut.edu.cn 
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Abstract:
      The stability of the basis is important for the computer manipulations. The hierarchical B-spline basis forms a convex partition of unity and has the smaller supports than those of the hierarchical B-spline basis. In this paper, we consider the stability of the truncated hierarchical B-spline basis respect to the $L_p$-norm, $\ 1\leq p < \infty$, We prove that this basis is weakly $L_p$-stable, $\ 1\leq p < \infty$, by using some classics tools of mathematical analysis. This means that the associated constant to be considered in the stability analysis is at most a polynomials growth in the number of the hierarchy depth.
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