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TitleStructure and dimension of multivariate spline space of lower degree on arbitrary triangulation
Author(s) Zhongxuan Luo, Ren-Hong Wang
TypeArticle in Journal
AbstractIn this paper, we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules. On the base of analyzing the algebraic and geometric results about singularity of S 2 1 ( Δ MS ) , we build the structure of triangulation and give some useful geometric conditions such that S μ + 1 μ ( Δ ) space is singular, and we obtain an algebraic condition which is necessary and sufficient for the singularity of S μ + 1 μ ( Δ ) spaces as well as their dimension formulae. Moreover, the structure matrix of spline spaces over any given partition is defined, which has been used to discuss the structure of S 3 1 ( Δ ) and S 2 1 ( Δ ) spaces over arbitrary triangulation and to prove the nonsingularity of S 3 1 ( Δ ) spaces. This partially settles a conjecture on the singularity of spline spaces in Wang et al., [Multivariate Spline and its Applications, Kluwer Press, Dordrecht, 2002; Academic Press, Beijing, 1994 (in Chinese)]. Meanwhile, the dimension formulae of S 3 1 ( Δ ) , S 2 1 ( Δ ) spaces and the dimension formulae of S μ + 1 μ ( Δ ) ( μ ⩾ 1 ) spaces are also given in this paper.
KeywordsMultivariate spline, Smoothing cofactor, Generator basis, Structure matrix
URL http://www.sciencedirect.com/science/article/pii/S0377042705004796
JournalJournal of Computational and Applied Mathematics
Pages113 - 133
NoteSpecial Issue: The International Symposium on Computing and Information (ISCI2004)
Translation No
Refereed No