{"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"}