Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials
Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials
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DOI：
10.1111/sapm.12387
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发表时间：
2020-08
影响因子：
2.7
通讯作者：
A. Barhoumi;Andrew F. Celsus;A. Deaño
中科院分区：
文献类型：
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作者：
A. Barhoumi;Andrew F. Celsus;A. Deaño
我们研究一族一族正交多项式，它们在区间 [−1,1] 上与变化的复值权重函数 exp(nsz) 正交，其中 s∈C 是任意的。该多项式族最初出现在文献中，当时参数为纯虚数，即 s∈iR ，因为它与高振荡积分的复杂高斯求积规则有关。最近，我们研究了 s∈iR 中 n→∞ 多项式的渐进性，我们的主要目标是将这些结果扩展到复平面中的所有 s。我们首先使用在可积系统理论背景下开发的参数空间连续技术，将先前关于所谓修正外场的结果从虚轴扩展到复平面减去一组临界曲线，称为断裂曲线。然后，我们应用 Deift 和 Zhou 在 1990 年代开发的振荡黎曼-希尔伯特问题的非线性最速下降方法，在参数 s 远离断裂曲线时获得这些多项式的递推系数的渐近性。然后，我们通过在 s 接近这些点时考虑双缩放限制，对参数 s 接近断裂曲线时的递归系数进行分析。我们看到递归系数的行为存在质的差异，具体取决于我们是否接近点 s=±2 或断裂曲线上的其他点。
We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp(nsz) , over the interval [−1,1] , where s∈C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s∈iR , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n→∞ have recently been studied for s∈iR , and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s=±2 or some other points on the breaking curve.