多割下的矩阵Szegő函数与矩阵正交多项式
Matrix Szegő Function and Matrix Orthogonal Polynomials for Multiple Cuts
AI总结：
本文针对多割下的矩阵权函数，构造其矩阵Szegő分解，通过Deift-Zhou最速下降分析得到矩阵值正交多项式的强渐近式，扩展了相关学者的已有研究。
AI中文摘要：
继矩阵值正交多项式（MVOPs）的最新进展，我们构造了支撑在多个区间上的矩阵权函数的矩阵Szegő分解，发现其在间隙上具有酉乘性跳跃。接下来，通过Deift-Zhou最速下降分析，研究与MVOPs的黎曼-希尔伯特问题相关的全局参量，将参量的行可视化为超椭圆黎曼曲面上向量丛的截面，构造了解并利用消失引理给出全局参量的另一种视角。作为应用，我们得到了多割下MVOPs的强渐近式，扩展了Deaño、Kuijlaars和Román的工作。
英文摘要：
Following recent developments on matrix-valued orthogonal polynomials (MVOPs), we construct the matrix Szegő factorisation of the matrix weight function when it is supported on multiple intervals. We find that on the gaps the matrix function has unitary multiplicative jumps. In the next part, performing the Deift-Zhou steepest descent analysis we study the associated global parametrix which comes from the Riemann-Hilbert problem of the MVOPs. A solution is constructed visualising the rows of the parametrix as sections of vector bundles on a hyper-elliptic Riemann surface. An alternate view point to the global parametrix is also presented using a vanishing lemma. As an application we obtain strong asymptotics of MVOPs for multiple cuts which extends the works of Deaño, Kuijlaars, and Román.
