On finitely generated closed ideals in \(H^{\infty}(D)\)
type: Item
instance of: scholarly article
title: On finitely generated closed ideals in \(H^{\infty}(D)\)
zbMATH Open document ID: 0564.46044
author: Jean Bourgain
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74693
review text: Assume \(f_ 1,...,f_ N\) 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 \(I(f_ 1,...,f_ N)\) generated by \(f_ 1,...,f_ N\), namely the property \[ | f(z)| \leq \alpha (| f_ 1(z)| +...+| f_ N(z)|)\quad for\quad z\in D \] for some function \(\alpha\) : \({\mathbb{R}}_+\to {\mathbb{R}}_+\) satisfying \(\lim_{t\to 0}\alpha (t)/t=0.\) 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 \[ | f(z)| \leq C\max_{1\leq j\leq N}| f_ j(z)| \quad for\quad z\in D \] for some constant C, does not necessary imply that f is in the closure of \(I(f_ 1,...,f_ N)\).
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 3899578
zbMATH Keywords: space of bounded analytic functions on the open unit disc
zbMATH Keywords: Carleson solution to the corona problem
MaRDI profile type: Publication
OpenAlex ID: W2327153070
cites work: Interpolations by bounded analytic functions and the corona problem
cites work: Q5618011
cites work: Approximation of harmonic functions
cites work: Q3922907
DOI: 10.5802/AIF.1032
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 3899578