The Hörmander--Bernhardsson extremal function

Abstract

We characterize the function $\ddot{I}†$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\ddot{I}€$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm \ddot{I}{„}_n$, $n=1,2,\dots$. Starting from the fact that $n+\frac{1}{2}-\ddot{I}{„}_n$ is an ${\ell }^2$ sequence, established in an earlier paper of ours, we identify $\ddot{I}†$ in the following way. We factor $\ddot{I}†(z)$ as $\hat{I}¦(z)\hat{I}¦(-z)$, where $\hat{I}¦(z)={\prod }_{n=1}^{\infty }(1+(-1{)}^n\frac{z}{\ddot{I}{„}_n})$ and show that $\hat{I}¦$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $\hat{I}¦$. We use these facts to establish an odd power series expansion of $n+\frac{1}{2}-\ddot{I}{„}_n$ in terms of $(n+\frac{1}{2}{)}^{-1}$ and a power series expansion of the Fourier transform of $\ddot{I}†$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $\hat{I}¦$ 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.

Some typos corrected. The paper will appear in Acta Mathematica