{"paper":{"title":"Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2011-12-21T00:17:58Z","abstract_excerpt":"We consider the space $A(\\mathbb T)$ of all continuous functions $f$ on the circle $\\mathbb T$ such that the sequence of Fourier coefficients $\\hat{f}=\\{\\hat{f}(k), ~k \\in \\mathbb Z\\}$ belongs to $l^1(\\mathbb Z)$. The norm on $A(\\mathbb T)$ is defined by $\\|f\\|_{A(\\mathbb T)}=\\|\\hat{f}\\|_{l^1(\\mathbb Z)}$. According to the known Beurling--Helson theorem, if $\\phi : \\mathbb T\\rightarrow\\mathbb T$ is a continuous mapping such that $\\|e^{in\\phi}\\|_{A(\\mathbb T)}=O(1), ~n\\in\\mathbb Z,$ then $\\phi$ is linear. It was conjectured by Kahane that the same conclusion about $\\phi$ is true under the assum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4892","kind":"arxiv","version":3},"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"}