We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function $e^{ni\lambda z}$ on $[-1,1]$, where $\lambda$ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when $\lambda$ is smaller than a certain critical value, ${\lambda }_c$. Our main goal is to compute their asymptotics when $\lambda >{\lambda }_c$.

We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann–Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift–Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.

Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of $\lambda$ for which one cannot solve the model Riemann–Hilbert problem, and as such the corresponding polynomials fail to exist.

我们研究了与变化的，高度振荡的复权函数正交的多项式族 ${Ë}^{ñ\mathrm{一世}\lambda ž}$ 上 $[-\mathrm{1个}，\mathrm{1个}]$，在哪里 $\lambda$是一个正参数。这个多项式族最近在文献中出现了与复杂的正交规则相关的问题，并且它们的渐近性以前已经在$\lambda$ 小于某个临界值， ${\lambda }_C$。我们的主要目标是计算它们的渐近性$\lambda >{\lambda }_C$。

我们首先根据二次微分理论对复平面中的曲线进行几何描述，这将最终支持这些多项式的渐近零分布。接下来，使用由Fokas，Its和Kitaev产生的正交多项式的强大Riemann-Hilbert公式，以及通过Deift-Zhou非线性最陡下降的渐近解方法，我们在整个复平面上提供了多项式的一致渐近性。

尽管这种渐近分析的大部分内容都遵循文献中以前的著作，但主要的障碍出现在所谓的全局参数的构造中。这种构造是借助于某些椭圆型积分以明确的方式进行的。与通常在实数正交的情况下遇到的情况形成鲜明对比的是，这种构造的一个有趣副产品是存在一组离散的$\lambda$ 对于这一点，无法解决模型的黎曼-希尔伯特问题，因此，相应的多项式就不存在了。