{"paper":{"title":"Factorizations of Schur functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OA","math.OC"],"primary_cat":"math.FA","authors_text":"Jaydeb Sarkar, Ramlal Debnath","submitted_at":"2019-08-05T20:42:22Z","abstract_excerpt":"The Schur class, denoted by $\\mathcal{S}(\\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\\mathbb{D}$ in the complex plane $\\mathbb{C}$, that is \\[ \\mathcal{S}(\\mathbb{D}) = \\{\\varphi \\in H^\\infty(\\mathbb{D}): \\|\\varphi\\|_{\\infty} := \\sup_{z \\in \\mathbb{D}} |\\varphi(z)| \\leq 1\\}. \\] The elements of $\\mathcal{S}(\\mathbb{D})$ are called Schur functions. A classical result going back to I. Schur states: A function $\\varphi: \\mathbb{D} \\rightarrow \\mathbb{C}$ is in $\\mathcal{S}(\\mathbb{D})$ if and only if there exist a Hilbert space $\\mathcal{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01850","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01850/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"}