{"paper":{"title":"A Note On Characterizations of Spherical t-Designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Congpei An","submitted_at":"2014-01-16T08:13:37Z","abstract_excerpt":"A set ${X}_{N}=\\{x_1,\\ldots,x_N\\}$ of $N$ points on the unit sphere $\\mathbb{S}^d,\\,d\\geq 2$ is a spherical $t$-design if the average of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial over ${X}_{N}$. This paper extends characterizations of spherical $t$-designs in previous paper from $\\mathbb{S}^2$ to general $\\mathbb{S}^d$. We show that for $N\\geq\\dim(\\mathbb{P}_{t+1})$, $X_N$ is a stationary point set of a certain non-negative quantity $A_{N,\\,t}$ and a fundamental system for polynomial space over $\\mathbb{S}^d$ with degree at most $t$, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3923","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"}