{"paper":{"title":"More on spherical designs of harmonic index $t$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Eiichi Bannai, Etsuko Bannai, Kyoung-Tark Kim, Wei-Hsuan Yu, Yan Zhu","submitted_at":"2015-07-20T02:53:54Z","abstract_excerpt":"A finite subset $Y$ on the unit sphere $S^{n-1} \\subseteq \\mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\\sum_{\\mathbf{x}\\in Y}f(\\mathbf{x})=0$ for all real homogeneous harmonic polynomials $f(x_1,\\ldots,x_n)$ of degree $t$. Also, for a subset $T$ of $\\mathbb{N} = \\{1,2,\\cdots \\}$, a finite subset $Y\\subset S^{n-1}$ is called a spherical design of harmonic index $T,$ if $\\sum_{\\mathbf{x}\\in Y}f(\\mathbf{x})=0$ is satisfied for all real homogeneous harmonic polynomials $f(x_1,\\ldots,x_n)$ of degree $k$ with $k\\in T$.\n  In the present p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05373","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"}