A Painlevé equation for the Hörmander-Bernhardsson constant

Abstract

The Hörmander-Bernhardsson constant $\mathcal{C}$ is the sharp constant in $|f(0)|\le \mathcal{C}\Vert f{\Vert }_1$ for entire functions of exponential type $\le \pi$. We prove that $\mathcal{C}=2\pi {\theta }_{\ast }^{-2}$ where ${\theta }_{\ast }$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Gordon equation $v_{\theta \theta }+v_{\theta }/\theta =\cosh (v)$. It is known that $\mathcal{C}$ 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.

28 pages, 1 figure. Version 2 adds Theorem 2 that proves the relation between $c_2$, $\mathcal{C}$, and $L_{\tau }(1)$ that was listed as a conjecture in the first version, and it provides a proof that the solutions of the RHP converge to the extremal functions when the Cosh-Gordon parameter approaches the singularity.Several notation clashes have been removed