{"paper":{"title":"Orthogonal polynomials and expansions for a family of weight functions in two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yuan Xu","submitted_at":"2010-12-23T18:08:55Z","abstract_excerpt":"Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \\CW_{\\a,\\b,\\g}(x,y) = |x+y|^{2\\a+1} |x-y|^{2\\b+1} (1-x^2)^\\g(1-y^2)^{\\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region bounded by two lines and a parabola. In the case of $\\g = \\pm 1/2$, an explicit basis of orthogonal polynomials is given in terms of Jacobi polynomials and a closed formula for the reproducing kernel is obtained. The latter is used to study the convergence of orthogonal expansions for these weight functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.5268","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":""},"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"}