Nearly outer functions as extreme points in punctured Hardy spaces

TL;DR

This paper characterizes the extreme points of the unit ball in a punctured Hardy space, extending classical results by identifying functions close to outer functions as these extreme points.

Contribution

It extends the classical characterization of extreme points in $H^1$ to punctured Hardy spaces with finitely many spectral holes.

Findings

Extreme points are functions close to being outer in $H^1_{ ext K}$.

Extension of de Leeuw and Rudin's theorem to punctured Hardy spaces.

Discussion of exposed points in the unit ball.

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 $\mathcal K$ of positive integers and consider the "punctured" Hardy space $$H^1_{\mathcal K}:=\{f\in H^1:\,\widehat f(k)=0\,\,\,\text{for all }\, k\in\mathcal K\}.$$ We then investigate the geometry of the unit ball in $H^1_{\mathcal K}$. In particular, the extreme points of the ball are identified as those unit-norm functions in $H^1_{\mathcal K}$ 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 $H^1_{\mathcal K}$.