数据几何与差商特征驱动的构造性逼近理论与方法研究

在数值逼近和计算几何的研究中，基函数一般通过节点序列来构造，逼近和插值表示式一般是直接对数据点进行构造。本项目基于数据的几何特征和差商特征研究逼近理论和方法，研究适应数据特征的基函数的表示，研究数据特征的直接表示。通过体现数据特征的基函数和数据特征的直接表示，构造数据的插值和逼近表示式，研究曲线和曲面的表示。研究数据特征的多项式表示、有理式表示、三角多项式表示和微分方程表示。研究从数据特征的各种表示模型转化到数据模型的求解方法。选择合适的目标准则和逼近式的类型，研究插值和逼近问题的最佳逼近解。研究曲面上数据特征约束的曲线生成的新方法。本项目的研究，对扩展逼近和插值中的数学问题的理论与方法，促进数值逼近与计算机辅助几何设计的理论和应用的发展，都具有重要的理论和应用意义。

On the numerical approximation and computational geometry, usually, the constructing of the basis functions is based on the nodal sequence, and the constructing of approximants and interpolants is made by using the given date. In this project, the study of the approximation theory and methods is based on the geometric and difference quotient characteristics of data. Based on the characteristics of data, we study the constructing of the basis functions and direct representation of data characteristics. By using the basis functions and the representation of data characteristics, we construct the approximants and interpolants for the data, and study the representations of curves and surfaces. With data characteristics, the representations of polynomials, rational functions, trigonometric polynomials and differential equations are studied. The methods for changing the models of data characteristics to data models are studied. By choosing the optimization criterions and function types, we study the solutions of best approximation for the problems of interpolation and approximation. With data characteristics constraint, new methods of generating curves on a surface are studied. The study on this project is of theory and applied significant on enriching the theory and methods of interpolation and approximation, and promoting the development of theory and application of numerical approximation and computer aided geometric design.