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On finitely generated closed ideals of $$H^ \infty$$. (Chinese) Zbl 0810.46058
Let $$\Omega$$ be a Denjoy domain and $$g, f_ 1,\dots, f_ n\in H^ \infty(\Omega)$$. Let $$\alpha(t)$$ be a nonnegative function on $$\mathbb{R}^ +$$ with $$\alpha(t)/t\to 0$$. If the following conditions are satisfied:
(1) $$| g(z)|\leq \alpha\left(\sum^ n_{j=1} | f_ j(z)|\right)$$, $$z\in \Omega$$;
(2) $$g(\bar z)= \overline{g(z)}$$, $$z\in \Omega$$;
(3) there exists $$\delta>0$$ such that $$| g(z)|\geq \delta$$, $$x\in \Omega\cap \mathbb{R}$$;
then $$g\in \overline{I(f_ 1,\dots, f_ n)}$$.
Here $$I(f_ 1,\dots, f_ n)$$ is the ideal generated by $$\{f_ 1,\dots, f_ n\}$$ and $$\overline{I(f_ 1,\dots, f_ n)}$$ is the closure of $$I(f_ 1,\dots, f_ n)$$.
##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
Denjoy domain