Variable degree polynomial splines are Chebyshev splines
Variable degree polynomial splines are Chebyshev splines
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DOI：
10.1007/s10444-011-9242-z
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发表时间：
2013-02
影响因子：
1.7
通讯作者：
Tina Bosner;M. Rogina
中科院分区：
文献类型：
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作者：
Tina Bosner;M. Rogina
变次多项式样条是一种很有价值的保形逼近方法。然而，人们期望样条空间的一些常见性质在几何建模中是有用的，并不容易从它们的定义中遵循。这包括总正性（TP）和变异减少，但也基于节点插入的建设性算法。本文考虑了在每个子区间上，i= 0，1，. L.本文的大部分内容都是关于非多项式情形的，而多项式样条（VDP-splines）是当m，n ∈ [4，∞）为整数时的特殊情形。我们把VDP-样条描述为一个正则完备Chebyshev函数系的分片张成，该函数系的测度向量由正有理函数sp（x），q（x）确定。这些功能是这样的，可变度样条属于piecewisely的核心的微分算子。虽然样条空间不是基于一个扩展的切比雪夫系统，我们认为，总的积极性和变化减少仍然成立。与抽象的结果，建设性的性质，如马斯登身份，准伯恩斯坦多项式和节点插入算法的递归性可能会涉及更多，我们证明他们只为VDP样条的订单4和5。
Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of orderspanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval,i= 0,1, ...l. Most of the paper deals with non-polynomial casemi,ni∈ [4, ∞ ), and polynomial splines known as VDP–splines are the special case whenmi,niare integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functionsp(x),q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator. Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.