The Hörmander--Bernhardsson extremal function

Abstract

We characterize the function $φ$ 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 $φ$ in the following way. We factor $φ(z)$ 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 $φ$, 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.