Nearly outer functions as extreme points in punctured Hardy spaces
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 $\mathscr{K}$ of positive i...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/217521 |
| Acceso en línea: | https://hdl.handle.net/2445/217521 |
| Access Level: | acceso abierto |
| Palabra clave: | Espais de Hardy Funcions de variables complexes Anàlisi harmònica Hardy spaces Functions of complex variables Harmonic analysis |
| Sumario: | 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 $\mathscr{K}$ of positive integers and consider the "punctured" Hardy space $$ H_{\mathscr{K}}^1:=\left\{f \in H^1: \widehat{f}(k)=0 \text { for all } k \in \mathscr{K}\right\} $$ We then investigate the geometry of the unit ball in $H_{\mathscr{X}}^1$. In particular, the extreme points of the ball are identified as those unit-norm functions in $H_{\mathscr{X}}^1$ 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 $H_{\mathscr{X}}^1$. |
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