Functions with small and large spectra as (non)extreme points in subspaces of $H^\infty$

Abstract:

Given a subset $\Lambda$ of $\mathbb Z_+≔\{0,1,2,\dots \}$, let $H^\infty (\Lambda )$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notin \Lambda$. Assuming that either $\Lambda$ or $\mathbb Z_+\setminus \Lambda$ is finite, we determine the extreme points of the unit ball in $H^\infty (\Lambda )$.