Research On The Existence Of (Vector-valued) Rational Interpolants

Rational function interpolation theory and its application are an important part in research on rational approximation. There have been a lot of achievements in uniqueness, 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 determinacy of these results will be influenced. The existing research on the existence of rational interpolation was carried out by means of Lagrange basis function or similar methods in which great calculation restricts 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 interpolants and osculatory rational interpolants by univariate and bivariate Newton assembly difference coefficient, 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 the judgment of the existence of rational interpolants.The chapter 3, we study the existence of univariate osculatory rational interpolants. We give a kind of algebraic method to judge the existence of univariate osculatory rational interpolants by use of univariate Newton interpolation polynomial and present a concrete expression of the corresponding rational interpolants when the latter exists.The chapter 4, we mainly discuss the existence of bivariate rational interpolants. At first, we present the method to judge the existence of bivariate vector valued rational interpolants ,then give the corresponding method for bivariate osculatory rational interpolants, after which we extend the above method to the judgment of the existence of bivariate vector valued osculatory rational interpolants and obtain corresponding result. In this part, the method is compact and practical. In other words, if rational intrpolants exists, the concrete expression can be presented and this method has some inheritance.