{"paper":{"title":"Sobolev orthogonal polynomials on the conic surface","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lidia Fernandez, Miguel Pinar, Teresa Perez, Yuan Xu","submitted_at":"2022-09-16T22:27:34Z","abstract_excerpt":"Orthogonal polynomials with respect to the weight function $w_{\\beta,\\gamma}(t) = t^\\beta (1-t)^\\gamma$, $\\gamma > -1$, on the conic surface $\\{(x,t): \\|x\\| = t, \\, x \\in \\mathbb{R}^d, \\, t \\le 1\\}$ are studied recently, and are shown to be eigenfunctions of a second order differential operator $\\mathcal{D}_\\gamma$ when $\\beta =-1$. We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th normal derivative $\\mathfrak{D} = \\frac{\\mathrm{d}}{\\mathrm{d} t} - t^{-1} \\langle x, \\nabla_x\\rangle$ of the cone with respect to $w_{\\beta+s,0}$ over the conic surface, p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.08186","kind":"arxiv","version":2},"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/2209.08186/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"}