On finitely generated closed ideals in $H^{\infty }\left(D\right)$

Bourgain, Jean. "On finitely generated closed ideals in $H^\infty (D)$." Annales de l'institut Fourier 35.4 (1985): 163-174. <http://eudml.org/doc/74693>.

@article{Bourgain1985,
abstract = {Assume $f_ 1,\ldots \{\},f_ N$ a finite set of functions in $H^\{\infty \}(D)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^\{\infty \}(D)$ to belong to the norm-closure of the ideal $I(f_ 1,\ldots \{\},f_ N)$ generated by $f_ 1,\ldots \{\},f_ N$, namely the property\begin\{\} \vert f(z)\vert \le \alpha (\vert f\_ 1(z)\vert +\ldots \{\}+\vert f\_ N(z)\vert )\ \text\{for\} \ z\in D \end\{\}for some function $\alpha $ : $\{\bf R\}_+\rightarrow \{\bf R\}_+$ satisfying $\lim _\{t\rightarrow 0\}\alpha (t)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property\begin\{\} \vert f(z)\vert \le C\max \_\{1\le j\le N\}\vert f\_ j(z)\vert \ \text\{for\} \ z\in D \end\{\}for some constant $C$, does not necessary imply that $f$ is in the closure of $I(f_ 1,\ldots \{\},f_ N)$.},
author = {Bourgain, Jean},
journal = {Annales de l'institut Fourier},
keywords = {space of bounded analytic functions on the open unit disc; Carleson solution to the corona problem},
language = {eng},
number = {4},
pages = {163-174},
publisher = {Association des Annales de l'Institut Fourier},
title = {On finitely generated closed ideals in $H^\infty (D)$},
url = {http://eudml.org/doc/74693},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Bourgain, Jean
TI - On finitely generated closed ideals in $H^\infty (D)$
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 4
SP - 163
EP - 174
AB - Assume $f_ 1,\ldots {},f_ N$ a finite set of functions in $H^{\infty }(D)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }(D)$ to belong to the norm-closure of the ideal $I(f_ 1,\ldots {},f_ N)$ generated by $f_ 1,\ldots {},f_ N$, namely the property\begin{} \vert f(z)\vert \le \alpha (\vert f_ 1(z)\vert +\ldots {}+\vert f_ N(z)\vert )\ \text{for} \ z\in D \end{}for some function $\alpha $ : ${\bf R}_+\rightarrow {\bf R}_+$ satisfying $\lim _{t\rightarrow 0}\alpha (t)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property\begin{} \vert f(z)\vert \le C\max _{1\le j\le N}\vert f_ j(z)\vert \ \text{for} \ z\in D \end{}for some constant $C$, does not necessary imply that $f$ is in the closure of $I(f_ 1,\ldots {},f_ N)$.
LA - eng
KW - space of bounded analytic functions on the open unit disc; Carleson solution to the corona problem
UR - http://eudml.org/doc/74693
ER -