Toeplitz矩阵是一类重要的结构矩阵, 在数学、物理学、工程学等领域有广泛的应用. 本文首先研究Toeplitz矩阵的有理生成函数问题, 给出这类矩阵可容许次数集合的刻画和具有给定可容许次数的所有有理生成函数的刻画. 然后, 应用Toeplitz矩阵的有理生成函数理论分别求解两点Padé逼近问题和Carathéodory-Fejer插值问题, 推导两点Padé逼近问题的逼近式、两点Padé逼近问题可解的充要条件和可解时解的表达式, 刻画两点Padé逼近表的块结构特征, 给出Carathéodory-Fejer插值问题的可解性准则和可解时解的表达式. 

全文共分为四章:

第一章是综述, 它包含三个部分. 第一部分介绍本文研究的主要问题、研究背景、研究方法和研究意义, 第二部分介绍本文的主要研究成果与创新点, 第三部分介绍本文的不足之处和后续研究工作.

在第二章, 我们通过引入Toeplitz矩阵的第一、第二特征度和特征多项式对, 给出Toeplitz矩阵的核结构特征、可容许次数集合和具有给定可容许次数的所有有理生成函数的刻画. 然后, 我们给出Toeplitz矩阵的拟直分解的概念, 证明非满秩Toeplitz矩阵的拟直分解的唯一性.

在第三章, 我们应用Toeplitz矩阵的有理生成函数理论研究两点Padé逼近问题, 给出两点Padé逼近问题的线性化问题的通解, 推导两点Padé逼近问题的逼近式、两点Padé逼近问题可解的充要条件和可解时解的表达式. 在此基础上, 我们研究两点Padé逼近表的块结构特征和表中相邻逼近式的分子、分母多项式之间的三项递推关系式.

第四章应用Hermite Toeplitz矩阵的有理生成函数理论研究Carathéodory-Fejer插值问题, 给出这类解析函数插值问题的可解性准则和可解时解的表达式.

Toeplitz matrices are an important class of structure matrices, and have numerous applications in many fields such as mathematics, physics, engineering, etc. In this paper, we first study the rational generating functions of a given Toeplitz matrix and describe the set of the admissible degrees of the Toeplitz matrix and the set of all rational generating functions of the Toeplitz matrix with a given admissible degree. Then we apply the theory of the rational generating functions of Toeplitz matrix to deal with the two-point Padé approximation problem and the Carathéodory-Fejer interpolation problem. We deduce the formula of the two-point Padé approximant, the necessary and sufficient conditions for the solvability of the two-point Padé approximation problem and the formula of the general solution when it is solvable, and give a description of the block structural characteristics of the two-point Padé table, the solvability criterion of the Carathéodory-Fejer interpolation problem and a formula of the solutions when it is solvable.

This paper is divided into four chapters:

Chapter 1 is an introduction of this paper, which includes three parts. The first part introduces the main problems considered in this paper, together with the method, background and significance of these problems. The second part introduces the main results and the innovations of this paper. The third part introduces the shortcomings of this paper and the further research on this subject.

In Chapter 2, we introduce the first and second characteristic degrees and a pair of characteristic polynomials of a given Toeplitz matrix. We describe the kernel structure characteristics and the set of the admissible degrees of the Toeplitz matrix，and give a description of all rational generating functions with a given admissible degree. Then, we give the definition of the quasi-direct decomposition of the Toeplitz matrix and prove the uniqueness of the quasi-direct decomposition of the Toeplitz matrix without full rank.

In Chapter 3, we study the two-point Padé approximation problem. By using the theory of the rational generating functions of Toeplitz matrix, we give a description of the solution set of the linearized problem, the formula of the two-point Padé approximant, the necessary and sufficient conditions for the solvability of the two-point Padé approximation problem and a description of the solution set when these conditions are met. In the rest of this chapter, we study the block structural characteristics of the two-point Padé table and deduce the three-term recursive relations for the numerators and denominators of three adjacent entries in the table.

Chapter 4 studies the Carathéodory-Fejer interpolation problem by using the theory of the rational generating functions of the Hermite Toeplitz matrix. The solvability criterion of the Carathéodory-Fejer interpolation problem is given and a formula of the solutions is deduced when it is solvable.