Nearly outer functions as extreme points in punctured Hardy spaces

Abstract

The Hardy space H1 consists of the integrable functions f on the unit circle whose Fourier coefficients fˆ(k) vanish for k<0. We are concerned with H1 functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set K of positive integers and consider the ‘‘punctured” Hardy spaceHK1:={f∈H1:fˆ(k)=0for all k∈K}. We then investigate the geometry of the unit ball in HK1. In particular, the extreme points of the ball are identified as those unit-norm functions in HK1 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 H1 and characterizes its extreme points as outer functions. We also discuss exposed points of the unit ball in HK1.