The Hörmander--Bernhardsson extremal function
Andriy Bondarenko, Joaquim Ortega-Cerdà, Danylo Radchenko, Kristian Seip
TL;DR
The paper resolves the Hörmander–Bernhardsson extremal problem by deriving dual characterizations of the extremal function, notably expressing $\varphi$ via $\Phi$ and establishing a second-order differential equation for $\Phi$ together with a functional equation that uniquely identifies $\Phi$ up to a scalar. It introduces a two-parameter family of differential operators $\mathcal{L}_{a,b}$ that commute with involutions, enabling a functional equation for eigenfunctions and a detailed spectral analysis that underpins high-precision computations of the constants $\mathscr{C}$ and $L_{\tau}(1)$. From these, the paper obtains a precise power-series description of the zeros $\tau_n$ through an odd function $\rho$, a sharp description of the Fourier transform of $\varphi$, and broad summation formulas linking $PW^1$-functions to the zero-set $\{ au_n\}$, together with numerics and several intriguing arithmetic connections. The work further connects the extremal problem to Dirichlet series $L_{\pm}(s)$ and proposes integrality phenomena, suggesting deeper algebraic structures and potential applications to time–frequency localization and analytic number theory.
Abstract
We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-τ_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $\varphi$ in the following way. We factor $\varphi(z)$ as $Φ(z)Φ(-z)$, where $Φ(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{τ_n})$ and show that $Φ$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $Φ$. We use these facts to establish an odd power series expansion of $n+\frac12-τ_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $\varphi$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $Φ$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.