{"paper":{"title":"Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feng Dai, Han Feng","submitted_at":"2017-05-13T17:55:01Z","abstract_excerpt":"This paper proves that given a doubling weight $w$ on the unit sphere $\\mathbb{S}^{d-1}$ of $\\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\\geq \\max_{x\\in \\mathbb{S}^{d-1}} \\frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\\cdots, z_N$ on $\\mathbb{S}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x) d\\sigma_d(x)$, $$ \\frac 1{w(\\mathbb{S}^{d-1})} \\int_{\\mathbb{S}^{d-1}} f(x) w(x)\\, d\\sigma_d(x)=\\frac 1N \\sum_{j=1}^N f(z_j),\\ \\ \\forall f\\in\\Pi_n^d, $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04864","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"}