{"paper":{"title":"Stabilization in $H^\\infty_{\\mathbb{R}}(\\mathbb{D})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Brett D. Wick","submitted_at":"2008-09-09T15:07:55Z","abstract_excerpt":"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 $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\\quad\\forall z\\in\\D. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.1573","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}