Some Problems In Approximation Theory With Radial Basis Function

The present Ph.D.dissertation is concerned with some problems in approximation theory with radial basis function.Radial basis function methods is one of the most popular methods in approximation theory.The advantage of radial basis function methods is solving the multivariate problems with scattered data easily as much as that in one dimension.It begins with radial basis function interpolation,and till now,it's well-developed except two problems: one is decomposition of the native space of radial basis function based on multi-resolution analysis;the other is the quasi-interpolation for scattered data with radial kernels.The present Ph.D.dissertation is studying about these problems.First,we generalized the sufficient and necessary conditions for nonstationary MRA tight frame to the general Hilbert functions space,and constructed nonstationary MRA tight frame of native space of radial basis functions.Second,we study the influence of shape parameter in quasi-interpolation which is proposed in[85],and then constructed a new quasi-interpolation with a new radial kernel.The convergence rate of new quasi-interpolation is fast than that in[85].And last, we show a series of numerical examples with different kinds of kernels and parameters. Several aspects have been studied,including the best choice of shape parameter and the numerical convergence rate.The whole dissretation is organized as follows.Chapter One is preface.This chapter is devoted to the background of these problems, and the main results of this Ph.D.dissertation are also illustrated.Chapter Two is preliminaries.We introduce some related knowledge which are needed in the dissertation.Chapter Three generalized the theory for nonstationary MRA tight frame to general Hilbert functions space,and got the sufficient and necessary conditions for nonstationary MRA tight frame of it.Chapter Four first introduced the basic theory of native space,and then constructed nonstationary MRA tight frame of native space of radial basis functions based on the results of Chapter Three,in other words,we gave the decomposition of the native space based on nonstationary MRA tight frame.Chapter Five first introduced the integral formula based on scattered data and the quasi-interpolation for multivariate scattered data which proposed in[85].and then we study the influence of shape parameter and get the error estimation with it.At the end of this chapter,we constructed a new radial kernel and then got a new quasi-interpolation for scattered data,which was shown to provide higher approximation order than that in [85].Chapter Six first gave a scheme to extend the function defined the so-called starlike compact domain to whole space with Hermitian extrapolation.Then we show a series of numerical examples with different kinds of kernels and parameters,and we study the best choice of shape parameter and the numerical convergence rate.