Some Problems In Approximation Theory With Spline Functions And Radial Basis Functions

Spline functions and radial basis functions are important tools in approximation theory and numerical analysis. In this dissertation, some problems in approximation theory with spline functions and radial basis functions are studied. The contents are summarized as follows:In Chapter 1, we present some preliminaries for the dissertation, including definitions of spline functions and radial basis functions, and some basic facts about them.In Chapter 2, we propose a multinomial spline approximation method by using spline quasi-interpolation. It can be considered as a refinement of the traditional Bernstein approximation. Error estimates and numerical examples show that this new method could produce high accurate results.In Chapter 3, we propose a kind of Bernoulli-type multiquadric quasi-interpolation operator by using Bernoulli polynomials, and give the approximation order of the operator. Moreover, we apply the operator to the fitting of discrete solutions of initial value problem, and obtain the results which are as accurate as those produced by Runge-Kutta method of order 4.In Chapter 4, we propose a numerical scheme for solving Burgers'equation using multi-level multiquadric quasi-interpolation. The associated algorithm is simple and easy to implemented. Numerical examples show that our scheme is more accurate than the schemes using multiquadric interpolation, multiquadric quasi-interpolation and finite el-ement method with moving nodes.In Chapter 5, we study the selection of the parameter in Lipschitz constant dimin-ishing interpolation for scattered data. We investigate numerically the influence of the parameter on the quality of approximation by Lipschitz constant diminishing interpolants, and propose an algorithm based on cross-validation, for selecting a good parameter. Nu-merical examples show that the algorithm always produces good values for the parameter.