{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:OOP7UFR2UY5QLYINPIQ6ZGQNLO","short_pith_number":"pith:OOP7UFR2","schema_version":"1.0","canonical_sha256":"739ffa163aa63b05e10d7a21ec9a0d5bb121f9b9f0a779a8101cf1e26509a912","source":{"kind":"arxiv","id":"1206.1526","version":1},"attestation_state":"computed","paper":{"title":"Fej\\'er-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Jeffrey S. Geronimo, Plamen Iliev","submitted_at":"2012-06-07T15:28:40Z","abstract_excerpt":"We give a complete characterization of the positive trigonometric polynomials Q(\\theta,\\phi) on the bi-circle, which can be factored as Q(\\theta,\\phi)=|p(e^{i\\theta},e^{i\\phi})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4\\pi^2Q(\\theta,\\phi)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.1526","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-06-07T15:28:40Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"b4c7dc22dfbc9cf918c5201ee0e578b15c74cc6e59627330735b97335f532caa","abstract_canon_sha256":"364478c1a0d5de40deca30c760b4bc30a4828b6ead0cb3ec56a675bd4cd15af3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:38.446082Z","signature_b64":"SSlKUDcsKtZVUDcV1D6MSkMjJhpQg7XGdg8hd1fK9V9MXCYnzuY8lFuiIPLPaeojnVGGOsw2MZXCHno8jlp0BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"739ffa163aa63b05e10d7a21ec9a0d5bb121f9b9f0a779a8101cf1e26509a912","last_reissued_at":"2026-05-18T02:39:38.445694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:38.445694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fej\\'er-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Jeffrey S. Geronimo, Plamen Iliev","submitted_at":"2012-06-07T15:28:40Z","abstract_excerpt":"We give a complete characterization of the positive trigonometric polynomials Q(\\theta,\\phi) on the bi-circle, which can be factored as Q(\\theta,\\phi)=|p(e^{i\\theta},e^{i\\phi})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4\\pi^2Q(\\theta,\\phi)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1526","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.1526","created_at":"2026-05-18T02:39:38.445754+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.1526v1","created_at":"2026-05-18T02:39:38.445754+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1526","created_at":"2026-05-18T02:39:38.445754+00:00"},{"alias_kind":"pith_short_12","alias_value":"OOP7UFR2UY5Q","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"OOP7UFR2UY5QLYIN","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"OOP7UFR2","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO","json":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO.json","graph_json":"https://pith.science/api/pith-number/OOP7UFR2UY5QLYINPIQ6ZGQNLO/graph.json","events_json":"https://pith.science/api/pith-number/OOP7UFR2UY5QLYINPIQ6ZGQNLO/events.json","paper":"https://pith.science/paper/OOP7UFR2"},"agent_actions":{"view_html":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO","download_json":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO.json","view_paper":"https://pith.science/paper/OOP7UFR2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.1526&json=true","fetch_graph":"https://pith.science/api/pith-number/OOP7UFR2UY5QLYINPIQ6ZGQNLO/graph.json","fetch_events":"https://pith.science/api/pith-number/OOP7UFR2UY5QLYINPIQ6ZGQNLO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO/action/storage_attestation","attest_author":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO/action/author_attestation","sign_citation":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO/action/citation_signature","submit_replication":"https://pith.science/pith/OOP7UFR2UY5QLYINPIQ6ZGQNLO/action/replication_record"}},"created_at":"2026-05-18T02:39:38.445754+00:00","updated_at":"2026-05-18T02:39:38.445754+00:00"}