Some Problem Research On Rational Function Approximants

Rational function interpolation theory and its application are an important part in research on rational approximation. There have been a lot of achievements in uniquenss, algorithms, error estimate and etc., especially in algorithms. But there doesn't always exist rational interpolation function for arbitrary interpolation conditions given in advance. Moreover, other results such as uniqueness, algorithms and error estimate are given which bases on that rational interpolation function exists . If the existence can't be settled well, the existence of rational interpolation was carried out by means of Lagrange basis function and Newton basis function or similar methods which great calculation restrict themselves application.In this thesis, we discuss how to judge the existence of the rational interpolation by use of geometric distribution of the type value points, and then give several methods for the judgment of the existence of the rational interpolations and osculatory rational interpolations by univariate generalized Vandermond determinant ,which are quick and practical.This thesis consists of four chapters. In chapter 1, we retrospect the background of the research on rational interpolants and the study actuality of the existence of rational interpolants.In chapter 2, we introduce two important results of rational interpolants, based on which we study the existence of rational interpolants by geometric distribution of points and give a intuitionistic method for judgment of the existence of rational interpolants.The chapter 3, we study the existence of univariate osculator rational interpolants. We give a kind of algebraic method to judge the existence of univariate osculatory rational interpolants by use of univariate generalized Vandermond determinant and present a concrete expression of the corresponding rational interpolants when the latter exists.The chapter 4, we mainly discuss the overlay algorithm of two-variable vector-valued rational interpolation and show formula of two-variable vector-valued contact interpolation. At first, we present the overlay algorithm of two-variable vector-valued rational interpolation by means of univariate Newton interpolation polynomial, then give the formula of two-variable vector-valued contact interpolation. In this part, the method is compact and practical. In other words, if rational interpolants exists, the concrete expression can be presented .