{"paper":{"title":"Weighted Sobolev orthogonal polynomials on the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Miguel A. Pinar, Teresa E. Perez, Yuan Xu","submitted_at":"2012-11-12T01:22:50Z","abstract_excerpt":"For the weight function $W_\\mu(x) = (1-|x|^2)^\\mu$, $\\mu > -1$, $\\lambda > 0$ and $b_\\mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$\n  \\la f,g \\ra = {b_\\mu [\\int_{\\BB^d} f(x) g(x) W_\\mu(x) dx +\n  \\lambda \\int_{\\BB^d} \\nabla f(x) \\cdot \\nabla g(x) W_\\mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $\\la \\cdot,\\cdot\\ra$, can be generated through a recursive formula."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2489","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"}