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A B-net Approach to the Study of Multivariate Splines

Cuo Zhurui and Jia Rongqing

(Zhejiang University

收稿日期: 1990-06-25
出版日期: 2015-05-15


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摘要  目前,在多元样条的研究工作中比较有效的三种方法是:B样条方法、光滑余因子方法及B网方法.关于B样条方法,作者之一在[1]中介绍了箱样条研究近年来的进展.关于光滑余因子方法,王仁宏作了介绍。关于B网方法,Farin作了一系列研究. 在本文中我们要介绍B网方法,侧重于与我们自身研究兴趣有关的一些问题.全文共分
Abstract:At present, three methods have proved successful in the study of multivariate splines. They are methods of B-splines, smoothing cofactors and B-nets. The third method, the B-net method, seems less known than the other two methods. This paper gives a discription of the B-net approach and a survey of the recent results in this field. No completeness has been desired. We are only concerned with the aspects of B-nets, in which we are most interested. It is hoped that this paper will make B-nets more popular.This paper divides into four sections. In 1, we introduce the B-net representations of polynomials on simplices and splines over triangulations. Bernstein polynomials on simplices are also discussed. We list the essential facts about B-net representation, in particular, we give a necessary and sufficient condition for a spline to be smooth in terms of its B-net representation .In 2 we employ the B-net method to study the algebraic properties of spaces of multivariate splines. It is demonstrated that many problems on dimension, basis. etc. can be solved successfully by a clever use of B-nets. In 3, we will show how the B-net method helps us to overcome the difficulty in determining the approximation order of bivariate spline spaces. In 4, from the view of B-nets, we summarize the various results on bivariate splines over a regular mesh.
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