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

Annales de l'institut Fourier (1985)

- Volume: 35, Issue: 4, page 163-174
- ISSN: 0373-0956

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topBourgain, 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 -

## References

top- [1] L. CARLESON, Interpolation of bounded analytic functions and the corona problem, Annals of Math., 76 (1962), 547-552. Zbl0112.29702MR25 #5186
- [2] P. DUREN, Theory of Hp-spaces, Academic Press, New York, 1970. Zbl0215.20203MR42 #3552
- [3] B. DAHLBERG, Approximation by harmonic functions, Ann. Inst. Fourier, Grenoble, 30-2 (1980), 97-101. Zbl0417.31005MR82i:31010
- [4] J. GARNETT, Bounded analytic functions, Academic Press, New York, 1980.

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