  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 set of extreme and
the set of exposed points of the unit ball of $PW^1_S$. The structure of these
sets becomes more complicated when $\rho<\sigma/2$.
