{"paper":{"title":"Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Edward B. Saff, Ian H. Sloan, Johann S. Brauchart, Josef Dick, Robert S. Womersley, Yu Guang Wang","submitted_at":"2014-07-31T08:37:28Z","abstract_excerpt":"We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \\subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\\frac{1}{N}\\sum_{\\mathbf{x} \\in X_N}f(\\mathbf{x}) \\approx \\int_{S^d} f \\, \\mathrm{d} \\sigma_d$ for functions in the Sobolev space $\\mathbb{W}_p^s(S^d)$, where $\\sigma_d$ denotes normalized area measure on $S^d.$ These bounds are close to optimal when $s$ is close to $d/p$. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.8311","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"}