积分型样条函数逼近新理论、新方法及应用研究

The purpose of this project is to research the integro spline approximation theory, methods and applications. The previous studies on this topic are very limited and need to be improved greatly. We will apply spline functions to study some new highly effective numerical methods for approximating a univariate or multivarite function and its derivatives or partial derivatives from the given integral values of successive subintervals or subdomains. Essentially, it is an approximation problem based on linear functionals. We will analyze the approximation theory and develop the numerical algorithms for the so-called integro interpolating splines, approximate integro interpolating splines, integro quasi-interpolating splines and approximate integro quasi-interpolating splines. Moreover, we will also study their practical applications in many fields. Our new theory and methods are greatly improved and have many advantages over the existing ones. (1) The new theory and methods will be applicable to non-periodic functions and will not need any additional boundary conditions or parameters; (2) The new integro splines will be able to approximate the function and its derivatives or partial derivatives with higher approximation ability; (3) The theory and methods of multivariate integro splines will be well studied; (4) The new algorithms will have lower computational complexity and be easy to implement; (5) The applications of our new theory and methods in numerical analysis, mathematical statistics and environmental science will be studied.

根据某一元或多元函数的积分值信息逼近此函数及其各阶导数或偏导数是一类基于线性泛函信息的函数逼近问题。本项目研究积分值型样条函数逼近问题的新理论、新方法及其应用，此方面已有的工作具有若干不足与局限，尚不够深入与完善。在本项目中，主要以样条函数为工具，充分利用给定的积分值数据，重点研究高效积分值型样条函数插值和拟插值以及近似插值和近似拟插值的逼近理论方法、误差估计、数值算法及实际应用。主要包括以下工作:（1）研究适用于一般非周期函数且不需要任何边界条件或参数的理论方法；（2）研究能够逼近函数本身以及若干阶导数或偏导数的高精度逼近理论方法；（3）深入开展研究多元积分型样条函数理论、性质、方法；（4）设计简洁高效、易于实现、计算量低的新算法；（5）研究新理论新方法在数值分析、数理统计、环境科学等领域的应用。

本项目研究积分型函数逼近问题，这是一类基于线性泛函信息的函数逼近问题。主要开展积分型样条函数逼近新理论方法的研究，即根据某函数的积分值信息去求得积分型样条函数进行函数及其导数逼近。为改进以往四次样条的导数逼近方法，项目组研究了一种新型的四次插值样条，构造了新型五阶导数的四次样条逼近格式，并进一步应用到了五阶常微分方程边值问题中。为克服若干已有方法的固有缺点，项目组研究了一种无边界条件的五次积分样条插值方法，应用该方法我们能够得到被逼近函数及其前五阶导数的高精度逼近。为改进二次积分样条的逼近效果，项目组也研究了一种不完全边界条件的五次积分样条插值方法，应用该方法我们也能够得到被逼近函数及其若干阶导数的高精度逼近。为进一步探究积分样条的逼近性质，项目组研究了四类二次积分插值样条函数在中节点所共有的超收敛性质，这些新超收敛性质说明了这些二次积分样条函数具有较高的逼近能力。此外，项目组其他相关方面也取得了若干成果。本项目所得结果在数值分析、数理统计、环境科学等领域中都有理论应用。