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Our starting point is a theorem of de Leeuw and Rudin that describes theextreme points of the unit ball in the Hardy space $H^1$. We extend this resultto 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--deLeeuw type theorem for the unit ball of $H^1_{\mathcal K}$, the space offunctions $f\in H^1$ whose Fourier coefficients $\widehat f(k)$ vanish for all$k\in\mathcal K$.

A Rudin--de Leeuw type theorem for functions with spectral gaps