{"paper":{"title":"Approximations in $L^1$ with convergent Fourier series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Martin Grigoryan, Michael Ruzhansky, Zhirayr Avetisyan","submitted_at":"2018-10-14T15:46:00Z","abstract_excerpt":"For a separable finite diffuse measure space $\\mathcal{M}$ and an orthonormal basis $\\{\\varphi_n\\}$ of $L^2(\\mathcal{M})$ consisting of bounded functions $\\varphi_n\\in L^\\infty(\\mathcal{M})$, we find a measurable subset $E\\subset\\mathcal{M}$ of arbitrarily small complement $|\\mathcal{M}\\setminus E|<\\epsilon$, such that every measurable function $f\\in L^1(\\mathcal{M})$ has an approximant $g\\in L^1(\\mathcal{M})$ with $g=f$ on $E$ and the Fourier series of $g$ converges to $g$, and a few further properties. The subset $E$ is universal in the sense that it does not depend on the function $f$ to be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06047","kind":"arxiv","version":1},"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"}