On finitely generated closed ideals in \(H^{\infty}(D)\)
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 \(I(f_ 1,...,f_ N)\).
space of bounded analytic functions on the open unit disc
scientific article; zbMATH DE number 3899578