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

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.