The Hörmander--Bernhardsson extremal function: A preliminary study

We study the function ${\phi }_1$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm {\tau }_n$, $n=1,2,\dots$, and the sequence $({\tau }_n-n-\frac{1}{2})$ has ${\mathrm{\ell }}^2$ norm bounded by $0.13$. The zeros ${\tau }_n$ can be computed by means of a fixed point iteration.