The Hörmander--Bernhardsson extremal function

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