霍尔曼德-伯恩哈德松常数的一个Painlevé方程
A Painlevé equation for the Hörmander-Bernhardsson constant
- North Dakota State University（北达科他州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究确定霍尔曼德-伯恩哈德松常数与cosh-Gordon方程正则解的奇点相关，通过将多项式问题转化为Padé逼近问题，结合Deift-Zhou最速下降法与Painlevé超越函数完成分析。
AI中文摘要：
霍尔曼德-伯恩哈德松常数$\nmathscr{C}$是指数型不超过$\n\pi$的整函数满足$|f(0)|\le \mathscr{C}\\|f\\|_1$的最佳常数。我们证明$\n\mathscr{C}=2\pi \theta_*^2$，其中$\n\theta_*$是cosh-Gordon方程$v_{\theta\theta}+v_\theta/\theta=\cosh(v)$在$v(0)=0$的正则解$v$的最小正奇点。已知$\n\mathscr{C}$是次数不超过$n$的多项式类似问题中$n$的标度极限，我们将该多项式问题重新表述为无穷远处的Padé逼近问题，通过Deift-Zhou最速下降法分析相关的矩阵黎曼-希尔伯特问题，其局部参数化由一个Painlevé超越函数构造。
英文摘要：
The Hörmander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le π$. We prove that $\mathscr{C} = 2πθ_*^{-2}$ where $θ_*$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Gordon equation $v_{θθ} + v_θ/θ= \cosh(v)$. It is known that $\mathscr{C}$ is a scaling limit in $n$ from the analogous problem for polynomials of degree $\le n$. We reformulate the polynomial problem as a Padé approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlevé transcendent.
