Extreme points of the unit ball of Paley-Wiener space over two symmetric intervals

Alexander Ulanovskii, Ilya Zlotnikov

Published 2021-08-18Version 1

Let $PW_S^1$ be the space of integrable functions on $\mathbb{R}$ whose Fourier transform vanishes outside $S$, where $S = [-\sigma,-\rho]\cup[\rho,\sigma]$, $0<\rho<\sigma$. In the case $\rho>\sigma/2$, we present a complete description of the extreme points of the unit ball of $PW_S^1$. This description is no longer true if $\rho<\sigma/2$. For $\rho>\sigma/2$ we also show that every $f \in PW^1_S, \, \|f\|_1 =1,$ can be represented as $f = (f_1 + f_2)/2$ where $f_1$ and $f_2$ are extreme.