索伯列夫正交多项式的黎曼-希尔伯特表示
A Riemann-Hilbert representation for Sobolev orthogonal polynomials
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中文总结 AI 辅助
本文将索伯列夫正交多项式表示为I型多重正交多项式，利用二阶常微分方程的WKB渐近性质建立其黎曼-希尔伯特问题，推导得到投影核的克里斯托费尔-达布型公式。
中文摘要 AI 辅助
本文研究关于内积的正交多项式，内积形式为$\boldsymbol{\rm \textless}P,Q\boldsymbol{\rm \textgreater}_S = \boldsymbol{\rm \textsmaller{R}}P(x)Q(x)\text{e}^{-V(x)}\text{d}x + \boldsymbol{\rm \textlambda}\boldsymbol{\rm \textsmaller{R}}P^\boldsymbol{\textprime}(x)Q^\boldsymbol{\textprime}(x)\text{e}^{-V(x)}\text{d}x$，其中$V$是次数至少为4的偶次多项式且首项系数为正，$\boldsymbol{\textlambda}>0$为常数。上述内积是所谓索伯列夫内积的特例。本文展示如何将相关的索伯列夫正交多项式表示为一类I型多重正交多项式，该对应关系利用了某二阶常微分方程的WKB渐近性质，由此可写出索伯列夫正交多项式的黎曼-希尔伯特问题，进而推导出投影核的克里斯托费尔-达布型公式。
英文摘要
In this article we consider polynomials orthogonal with respect to the inner product $$\langle P,Q \rangle_S = \int_{\mathbb{R}}P(x)Q(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x + λ\int_{\mathbb{R}}P^\prime(x)Q^\prime(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and positive leading coefficient, and $λ> 0$ a constant. The above inner product is a special case of the so-called Sobolev inner product. We show how one can represent the associated Sobolev orthogonal polynomials as a species of Type I multiple-orthogonal polynomial. This correspondence makes use of the WKB asymptotics of a certain second order ODE. From this we may write a Riemann--Hilbert problem for the Sobolev orthogonal polynomials, from which one can deduce a Christoffel--Darboux-type formula for the projection kernel.
