{"paper":{"title":"Revisiting mean-square approximation by polynomials in the unit disk","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Bartosz Malman","submitted_at":"2023-04-03T22:33:38Z","abstract_excerpt":"For a positive finite Borel measure $\\mu$ compactly supported in the complex plane, the space $\\mathcal{P}^2(\\mu)$ is the closure of the analytic polynomials in the Lebesgue space $L^2(\\mu)$. According to Thomson's famous result, any space $\\mathcal{P}^2(\\mu)$ decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual $L^2$-space. We study the structure of this decomposition for a class of Borel measures $\\mu$ supported on the closed unit disk for which the part $\\mu_\\mathbb{D}$, living in the open disk $\\mathbb{D}$, is radial and decreases at least exponentially "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.01400","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.01400/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}