A Painlev\'e equation for the H\"ormander-Bernhardsson constant

Abstract

The H\"ormander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le \pi$. We prove that $\mathscr{C} = 2\pi \theta_*^{-2}$ where $\theta_*$ 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 $\mathscr{C}$ is a scaling limit in $n$ from the analogous problem for polynomials of degree $\le n$. We reformulate the polynomial problem as a Pad\'e 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 Painlev\'e transcendent.