基于逼近型细分的诱导细分格式

Subdivision schemes induced by approximating schemes

利用逼近型细分构造插值型细分是细分领域中的一个重要问题,目前可以给出插值型细分生成函数的研究还非常少.本文给出一个生成函数的统一公式,该公式由逼近型细分的生成函数与一个子生成函数构成.该公式对应一个插值型细分或者逼近型细分,这个取决于子生成函数的选取.该公式在理论和实际中都很重要.首先,这个公式适用于任意伸缩矩阵的多元基本型细分;其次,不论是一元细分还是多元细分,推导这个统一公式都不需要求解线性方程组;再次,这个公式具有显著的几何意义,应用方便;最后,从理论上分析诱导细分的零条件和多项式再生性,本文发现这些性质不仅与逼近型细分的零条件有关,而且与逼近型细分的多项式再生性有关,从而对细分格式的构造有指导意义.本文给出3个例子来说明这个统一公式.

Construction of new interpolatory subdivision schemes from approximating subdivision schemes is a hot topic that recently emerges in the field of subdivision method. However, rather less attention has been paid to the unified formula that can express the interpolatory subdivision generating function analytically in terms of the approximating subdivision symbol. This paper presents such a unified subdivision generating function formula that consists the generating function of the approximating subdivision scheme and one subsymbol. This unified subdivision generating function formula can be a generating function of an interpolatory scheme or that of an approximating scheme depending on the specific choice of the subsymbol. This formula plays an important role in theories and applications. First, it can be applied to higher-dimensional subdivisions with arbitrary dilation matrix, and it can also induce approximating schemes. Second, it does not need to solve a system of linear equations. Third, this formula has significant geometric meanings. At last, the specific structure of the formula also implies that zero condition properties of new subdivision scheme is closely related with the properties of the primal approximating subdivision. We illustrate the unified formula with three examples.