论文标题
$ [-1,1] $的矩阵矩阵的渐近差异分数
Asymptotics of matrix valued orthogonal polynomials on $[-1,1]$
论文作者
论文摘要
我们分析了矩阵有价值的正交多项式（MVOPS）的大渐近行为，其权重由雅各比标量因子和基质部分组成。使用Riemann-Hilbert配方进行MVOPS和Deift-Zhou最陡后的Deift-Zhou方法，我们在复杂平面的不同区域（在正交性的不同区域（在正交间隔之外），在远离端口和范围内的近距离范围内，在远处的不同区域内，在这些端口的不同区域中获得了MVOPS的渐近扩展，以及三个中间的coefferix intrix coefferix intrix intrix intrix intrix intrix intrix。 mvops。渐近分析遵循Kuijlaars，McLaughlin，Van Assche和Vanlessen在标量jacobi-type正交多项式上的工作，但就特征值/eigenValues/eigenvectors和eigenvectors及其使用MattrixSzegősegrixSzeg®功能而言，它也需要对矩阵重量部分的几个不同的因素化。我们以两个主要示例（雅各布和吉根鲍尔类型的Mvops）来说明结果，来自群体理论。
We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann-Hilbert formulation for MVOPs and the Deift-Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
