Soft Riemann-Hilbert problems and planar orthogonal polynomials

IF 4.3 3区 材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC
Haakan Hedenmalm
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引用次数: 0

Abstract

Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to exponentially varying weights in the plane, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), based on the technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a certain matrix ¯ $\bar{\partial }$ -problem (Its, Takhtajan), which we refer to as a soft Riemann-Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann-Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann-Hilbert problems along the interface (the free boundary for a related obstacle problem). Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real-analytic weights, the asymptotics holds in the L2 sense with error O ( e δ m ) $\mathrm{O}(\mathrm{e}^{-\delta \sqrt {m}})$ in a fixed neighborhood of the closed exterior of the interface, for some constant δ > 0 $\delta &gt;0$ , where m + $m\rightarrow +\infty$ . Here, m is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface (by a distance proportional to m 1 4 $m^{-\frac{1}{4}}$ ). In particular, the zeros of the orthogonal polynomial are located in the interior of the spectral droplet, away from the droplet boundary by a distance at least proportional to m 1 4 $m^{-\frac{1}{4}}$ .

软Riemann-Hilbert问题与平面正交多项式
Riemann-Hilbert问题是全纯函数在给定界面上的跳跃问题。它们出现在各种情况下,例如,在某些非线性偏微分方程的渐近研究和正交多项式的渐近分析中。Deift等人考虑了矩阵值的Riemann-Hilbert问题。在20世纪90年代,对最速下降法进行了非对易改编。对于线上或圆上关于指数变化权重的正交多项式,这导致给定参数的强渐近展开。对于关于平面中指数变化权重的正交多项式,Hedenmalm和Wennman(2017)基于正交性的不变叶理的技术构建获得了相应的渐近性。平面正交多项式的特征在于一个特定的矩阵?$\bar{\partial}$-问题(Its,Takhtajan),我们称之为软黎曼-希尔伯特问题。在这里,我们使用这个观点来提供一种简化的方法,该方法不是基于叶理,而是基于软黎曼-希尔伯特问题中Cauchy势的代数变换的特设插入。这允许问题沿着界面(相关障碍物问题的自由边界)分解为标量黎曼-希尔伯特问题的层次。受微观局部分析的启发,该方法允许以这样一种方式控制解,即对于真实的分析权重,渐近性在L2意义上成立,误差为O(e-δm)$\mathrm{O}(\mathrm{e}^{-\delta\sqrt{m})$,在界面闭合外部的固定邻域中,对于某个常数δ>;0$\delta>;0$,其中m→+∞$m\rightarrow+\infty$。这里,m是多项式的次数,就逐点渐近性而言,扩展在外域和界面上的误差项中占主导地位(与m−14$m^-\frac{1}{4}}$成比例的距离)。特别地,正交多项式的零点位于光谱液滴的内部,与液滴边界相距至少与m−14$m^-\frac{1}{4}}$成比例的距离。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
7.20
自引率
4.30%
发文量
567
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