论文标题
具有真实极点，根渐近肌和GMP矩阵的正交合理函数
Orthogonal rational functions with real poles, root asymptotics, and GMP matrices
论文作者
论文摘要
关于$ \ mathbb {r} $的度量及其在雅各比矩阵上的应用，正交多项式的渐近性行为的渐近行为具有广泛的理论。该理论具有明显的仿射不变性，并且对于$ \ infty $来说是非常特殊的角色。我们将该理论的各个方面扩展在$ \ Overline {\ Mathbb {r}} = \ Mathbb {r} \ cup \ {\ infty \} $上的有理函数的设置中，获得了多个电池，并证明了$ \ operline的不变性，并获得了$ \ mathbb的不变性。我们从其矩阵元素方面获得了GMP矩阵的stahl--totik规律性的表征；作为应用程序，我们给出了Simon的猜想的证明 - 有限间隙集上常规雅各比矩阵的nevai属性。
There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving Möbius transformations. We obtain a characterization of Stahl--Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon -- a Cesàro--Nevai property of regular Jacobi matrices on finite gap sets.
