| $L_p$ Stability of the Truncated Hierarchical B-Spline Basis |
| Received:May 25, 2017 Revised:September 01, 2017 |
| Key Words:
hierarchical spline space truncated hierarchical B-spline basis stable basis condition number of basis partition of unity
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| Fund Project: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). |
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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2017.06.006 |
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