任意剖分下的多元样条分析

本文采用代数几何的方法,研究了在任意剖分下多元样条函数的各种性质.定理2—4给出了一个函数S(υ,ν)是多元参数型样条的充分必要条件.定理1指出了多元样条函数具有“解析延拓”的特征性质.文中得到在任意剖分下多元样条的一般表达形式(定理9和10)和多元样条插值的一般理论.文中也讨论了多元有理样条函数.

ON THE ANALYSIS OF MULTIVARIATE SPLINES IN THE CASE OF ARBITRARY PARTITION

This paper aims at employing algebraic-geometric methods to investigate multivariate parametric splines in the case of arbitrary partition. The Theorems 2, 3 and 4 give necessary and sufficient conditions for a function S(u, v) to be a multivariate parametric spline. Theorem 1 is proved to have the property that the multivariate splines are characterized by a certain "analytic extension". In §5, the general rcpresentative formulas for multivariate splines are obtained in the case of arbitrary partition (Theorems 9 and 10). In §7, the general theory of interpolation is established for multivariate splines. Moreovcr, the multivariate rational splines are discussed.