{"paper":{"title":"Bernstein-Szeg\\H{o} measures in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Bernstein-Szegő measures on R² are defined via a new identity linking Fejér-Riesz factorization of the weight to a three-variable polynomial, yielding explicit orthonormal bases and complete characterization by finitely many moments.","cross_cats":["math.CV","math.FA"],"primary_cat":"math.CA","authors_text":"Jeffrey S. Geronimo, Plamen Iliev","submitted_at":"2022-07-28T21:25:52Z","abstract_excerpt":"We define a class of Bernstein-Szeg\\H{o} measures on $\\mathbb{R}^2$ and we establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures. A key ingredient in the theory on the real line stems from the fact that a measure $\\mu$ on $\\mathbb{R}$ determines a unique sequence of orthonormal polynomials which gives a simple formula for $d\\mu/dx $ in the Bernstein-Szeg\\H{o} family. Since there is no canonical way "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We define a class of Bernstein-Szegő measures on R² and establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a new identity exists connecting a Fejér-Riesz factorization of the weight to a polynomial in three variables associated with the measure, and that recent bivariate trigonometric Fejér-Riesz results suffice to define a nontrivial two-dimensional Szegő mapping yielding explicit orthonormal bases.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Defines Bernstein-Szegő measures on R², derives new finite-moment characterization conditions, and constructs orthonormal bases via an extended Szegő mapping from bivariate Fejér-Riesz factorization.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Bernstein-Szegő measures on R² are defined via a new identity linking Fejér-Riesz factorization of the weight to a three-variable polynomial, yielding explicit orthonormal bases and complete characterization by finitely many moments.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"57a071ecedb3807cdb779eaa78431bd6452e6ab097d78176830f192793c30185"},"source":{"id":"2207.14383","kind":"arxiv","version":4},"verdict":{"id":"9facdf64-9510-4464-a794-17acee7518c9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T11:42:39.164979Z","strongest_claim":"We define a class of Bernstein-Szegő measures on R² and establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures.","one_line_summary":"Defines Bernstein-Szegő measures on R², derives new finite-moment characterization conditions, and constructs orthonormal bases via an extended Szegő mapping from bivariate Fejér-Riesz factorization.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a new identity exists connecting a Fejér-Riesz factorization of the weight to a polynomial in three variables associated with the measure, and that recent bivariate trigonometric Fejér-Riesz results suffice to define a nontrivial two-dimensional Szegő mapping yielding explicit orthonormal bases.","pith_extraction_headline":"Bernstein-Szegő measures on R² are defined via a new identity linking Fejér-Riesz factorization of the weight to a three-variable polynomial, yielding explicit orthonormal bases and complete characterization by finitely many moments."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.14383/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":28,"sample":[{"doi":"","year":1968,"title":"Ju. M. Berezans ′ki ˘ ı,Expansions in eigenfunctions of selfadjoint operators , Translations of Mathematical Monographs, Vol. 17, American Mathematical S ociety, Providence, R.I., 1968","work_id":"914c33a2-8edd-4d79-9408-8e6606cba792","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"D. Damanik, A. Pushnitski and B. Simon, The analytic theory of matrix orthogonal polyno- mials, Surv. Approx. Theory 4 (2008), 1–85","work_id":"56f8fcbd-1b0c-4019-a8eb-4fb34b34523f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"D. Damanik and B. Simon, Jost functions and Jost solutions for Jacobi matrices. II. D ecay and analyticity , Int. Math. Res. Not. 2006, Art. ID 19396, 32 pp","work_id":"4bc169be-e61e-42fd-8d63-3903732ccb63","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"A. Delgado, J. Geronimo, P. Iliev and F. Marcell´ an, Two variable orthogonal polynomials and structured matrices , SIAM J. Matr. Anal. Appl. 28 (2006), no. 1, 118–147","work_id":"121f72c7-09a4-41d0-ae18-8ffa217b0125","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"A. Delgado, J. Geronimo, P. Iliev and Y. Xu, On a two variable class of Bernstein-Szeg˝ o measures, Constr. Approx. 30 (2009), no. 1, 71–91","work_id":"3474d539-bdbe-4992-890e-96103885b3eb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":28,"snapshot_sha256":"6d7fa7ecf1f4d761a42337f3bc81e540f7d3fa438d545b6cf643597995a609c6","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"}