# ANNALES DE L'INSTITUT FOURIER

On finitely generated closed ideals in ${H}^{\infty }\left(D\right)$
Annales de l'Institut Fourier, Tome 35 (1985) no. 4, p. 163-174
Soit ${f}_{1},...,{f}_{N}$ un nombre fini de fonctions dans l’espace ${H}^{\infty }\left(D\right)$ des fonctions bornées analytiques dans le disque unité ouvert $D$. Nous donnons une condition suffisante pour qu’une fonction $f$ dans ${H}^{\infty }\left(D\right)$ appartienne à l’adhérence pour la norme de l’idéal $I\left({f}_{1},...,{f}_{N}\right)$ engendré par ${f}_{1},...,{f}_{N}$, notamment la propriété$|f\left(z\right)|\le \alpha \left(|{f}_{1}\left(z\right)|+...+|{f}_{N}\left(z\right)|\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{pour}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in D$$\alpha$ est une fonction sur ${\mathbf{R}}_{+}$ telle que ${lim}_{t\to 0}\frac{\alpha \left(t\right)}{t}=0$. Le point essentiel est une amélioration dans la construction du contour, due à L. Carleson, liée au théorème de la couronne. Il est démontré aussi que la propriété$|f\left(z\right)|\le C\underset{1\le j\le N}{max}|{f}_{j}\left(z\right)|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{pour}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in D$pour une constante $C$, n’entraîne pas nécessairement que $f$ est dans l’adhérence de $I\left({f}_{1},...,{f}_{N}\right)$.
Assume ${f}_{1},...,{f}_{N}$ a finite set of functions in ${H}^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in ${H}^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I\left({f}_{1},...,{f}_{N}\right)$ generated by ${f}_{1},...,{f}_{N}$, namely the property$|f\left(z\right)|\le \alpha \left(|{f}_{1}\left(z\right)|+...+|{f}_{N}\left(z\right)|\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in D$for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ 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$|f\left(z\right)|\le C\underset{1\le j\le N}{max}|{f}_{j}\left(z\right)|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in D$for some constant $C$, does not necessary imply that $f$ is in the closure of $I\left({f}_{1},...,{f}_{N}\right)$.
@article{AIF_1985__35_4_163_0,
author = {Bourgain, Jean},
title = {On finitely generated closed ideals in $H^\infty (D)$},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
address = {28 - Luisant},
volume = {35},
number = {4},
year = {1985},
pages = {163-174},
doi = {10.5802/aif.1032},
mrnumber = {87j:46101},
zbl = {0564.46044},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_1985__35_4_163_0}
}

Bourgain, Jean. On finitely generated closed ideals in $H^\infty (D)$. Annales de l'Institut Fourier, Tome 35 (1985) no. 4, pp. 163-174. doi : 10.5802/aif.1032. https://aif.centre-mersenne.org/item/AIF_1985__35_4_163_0/

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