Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates.
翻译:矢量子配置办法使用矩阵面罩,是一种快速迭代平均算法,用于计算数字式 PDE 中波盘方法的可重新确定矢量函数,并在 CAGD 中生成平稳曲线。与经过仔细研究的 卡拉拉亚子配置办法相比,矢量子配置办法远未很好理解,例如,Lagrange 和(一般化的)Hermite 亚配置办法,是文献中唯一研究过的矢量子配置办法。由于数字式 PDE 中使用的许多波子来自可重新定义的矢量函数,其矩阵面罩并非来自 Hermite 子配置办法,因此有必要引入和研究任何通用矩阵面罩的矢量子配置办法,以便高效率地配置波子配置和可重新确定矢量配置办法的功能。对于一般矩阵子配置办法而言,只有一种有意义的方法来确定矢量源子配置办法。受矢量源源级演算算法和最近关于Hermite 子配置办法的研究,我们将为任何任意性矩阵面罩的矢量分组组合办法的矢量分配置办法确定矢量组合办法的矢量组合办法,但我们证明新定义的矢量分组组合的矢量组合的矢量组合的矢量组合安排的矢量组合安排的矢量组合安排的结果,也没有加强其现有矢量级次组合的矢量组合的矢量组合的矢量组合的矢量组合的矢量安排的矢量安排的矢量组合安排的矢量安排的矢量组合结果。