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

TL;DR

This paper characterizes the extreme points of the unit ball in certain subspaces of bounded analytic functions, focusing on those with either small or large spectra, under finiteness conditions.

Contribution

It provides a complete description of the extreme points in $H^inity(\Lambda)$ when $\Lambda$ or its complement is finite, extending understanding of the geometric structure of these function spaces.

Findings

Identifies extreme points for spaces with finite spectra.

Characterizes functions with small and large spectra as (non)extreme points.

Enhances geometric understanding of subspaces of $H^$.

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 $H^\infty(\Lambda)$.