The Hörmander--Bernhardsson extremal function: A preliminary study

Abstract

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

Title changed. To appear in a volume honoring Ed Saff, Appl. Numer. Harmon. Anal