埃尔米特线性泛函和差分方程的多项式扰动
Polynomial perturbations of hermitian linear functionals and difference equations
摘要：
本文研究了单位圆上的矩泛函的一般（Laurent）多项式修正，与Hermitian Toeplitz矩阵相关。我们提出了一种新的方法，使我们能够研究多项式修改的任意程度。主要目的是表征的拟定性的泛函中涉及的问题，在一个差分方程有关相应的Schur参数。结果是在一般框架（不一定准定）厄米泛函，使最大数量的正交多项式的特点是一致的步骤的算法的数量的基础上提到的递归的Schur参数。单位圆上正交多项式研究的一些具体应用表明了这种新方法的有效性：对勒贝格测度的一般一次逆多项式扰动所产生的泛函进行了详尽而有指导意义的分析，对多项式系数为常数的一类线性关系所连接的正交多项式对进行了分类，并给出了一些新的结果。以及确定其相关多项式与原始正交性泛函的一次多项式修改相关的那些正交多项式。
英文摘要：
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a kind of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.
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