{"paper":{"title":"Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.OC"],"primary_cat":"math.NA","authors_text":"Cordian Riener, Markus Schweighofer","submitted_at":"2016-07-28T11:13:21Z","abstract_excerpt":"Let $d$ and $k$ be positive integers. Let $\\mu$ be a positive Borel measure on $\\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. If the support of $\\mu$ is contained in an algebraic curve of degree $k$, then we show that there exists a quadrature rule for $\\mu$ with at most $dk$ many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most $2d-1$. This generalizes both Gauss and (the odd degree case of) Szeg\\H{o} quadrature where the curve is a line and a circle, respectively, to arbitrary plane algebraic curves. We use this result to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08404","kind":"arxiv","version":3},"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"}