Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$

Details Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$ Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$

2008-09-09

arxiv.org/abs/0809.1573v2

Publ. Mat. 54 (2010), no. 1, 25-52

Mathematics

Complex Variables

v1: 22 pages, 2 figures, to appear in Pub. Mat; v2: 32 pages, 5 figures. The earlier version incorrectly claimed a characteriz

Scientific paper

In this paper we prove the following theorem: Suppose that $f_1,f_2\in H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. $$ Assume for some $\epsilon>0$ and small, $f_1$ is positive on the set of $x\in(-1,1)$ where $\abs{f_2(x)}<\epsilon$ for some $\epsilon>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\in H^\infty_\R(\D)$ with $$ \norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq C(\delta,\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. $$

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