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 $H_{\mathcal{K}}^1$, the space of functions $f\in H^1$ whose Fourier coefficients $\widehat{f}(k)$ vanish for all $k\in \mathcal{K}$.