The H\"ormander--Bernhardsson extremal function: A preliminary study

TL;DR

This paper investigates the properties of a specific extremal function of minimal L^1 norm with exponential type constraints, focusing on its zeros and computational methods, building on earlier work by H"ormander and Bernhardsson.

Contribution

It provides a detailed analysis of the extremal function's zeros, their distribution, and introduces a fixed point iteration for their computation.

Findings

Zeros are real and symmetric about zero.

The sequence of zeros has bounded deviation from integers.

A fixed point iteration effectively computes the zeros.

Abstract

We study the function $\varphi_1$ 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 H\"{o}rmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$, and the sequence $(\tau_n-n-\frac12)$ has $\ell^2$ norm bounded by $0.13$. The zeros $\tau_n$ can be computed by means of a fixed point iteration.