Nearly outer functions as extreme points in punctured Hardy spaces

Abstract

The Hardy space $H^1$ consists of the integrable functions $f$ on the unit circle whose Fourier coefficients $\widehat{f}(k)$ vanish for $k<0$. We are concerned with $H^1$ functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set $\mathscr{K}$ of positive integers and consider the "punctured" Hardy space $$ H_{\mathscr{K}}^1:=\left\{f \in H^1: \widehat{f}(k)=0 \text { for all } k \in \mathscr{K}\right\} $$ We then investigate the geometry of the unit ball in $H_{\mathscr{X}}^1$. In particular, the extreme points of the ball are identified as those unit-norm functions in $H_{\mathscr{X}}^1$ which are not too far from being outer (in the appropriate sense). This extends a theorem of de Leeuw and Rudin that deals with the classical $H^1$ and characterizes its extreme points as outer functions. We also discuss exposed points of the unit ball in $H_{\mathscr{X}}^1$.