Nearly outer functions as extreme points in punctured Hardy spaces

The Hardy space H consists of the integrable functions f on the unit circle whose Fourier coefficients f̂(k) vanish for k < 0. We are concerned with H 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 space H K := {f ∈ H 1 : f̂(k) = 0 for all k ∈ K}. We then investigate the geometry of the unit ball inH K . In particular, the extreme points of the ball are identified as those unit-norm functions in H 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 and characterizes its extreme points as outer functions. We also discuss exposed points of the unit ball in H K .