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 $\hat{f}(k)$ 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 $$H_{\mathcal{K}}^1:=\{f\in H^1:\hat{f}(k)=0\text{for all }k\in \mathcal{K}\}.$$ We then investigate the geometry of the unit ball in $H_{\mathcal{K}}^1$. In particular, the extreme points of the ball are identified as those unit-norm functions in $H_{\mathcal{K}}^1$ 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 $H_{\mathcal{K}}^1$.

19 pages

References in corpus (2)

• Lacunary polynomials in $L^1$: geometry of the unit sphere
• A Rudin--de Leeuw type theorem for functions with spectral gaps

Cited by in corpus (4)

• A Rudin--de Leeuw type theorem for functions with spectral gaps
• Inner functions as strongly extreme points: stability properties
• On geometry of the unit ball of Paley-Wiener space over two symmetric intervals
• Functions with small and large spectra as (non)extreme points in subspaces of $H^{\infty }$