  Our starting point is a theorem of de Leeuw and Rudin that describes the
extreme points of the unit ball in the Hardy space $H^1$. We extend this result
to subspaces of $H^1$ formed by functions with smaller spectra. More precisely,
given a finite set $\mathcal K$ of positive integers, we prove a Rudin--de
Leeuw type theorem for the unit ball of $H^1_{\mathcal K}$, the space of
functions $f\in H^1$ whose Fourier coefficients $\widehat f(k)$ vanish for all
$k\in\mathcal K$.
