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

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 $k\in\mathcal K$.