{"paper":{"title":"A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"J. Brauchart, J. Dick","submitted_at":"2012-03-23T02:30:21Z","abstract_excerpt":"In this paper we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\\mathbb{S}^d \\subset \\mathbb{R}^{d+1}$. The reproducing kernel is given by an integral representation using the truncated power function $(\\boldsymbol{x} \\cdot \\boldsymbol{z} - t)_+^{\\beta-1}$ defined on spherical caps centered at $\\boldsymbol{z}$ of height $t$, which reduce to an integral over indicator functions of spherical caps as studied in [J. Brauchart, J. Dick, arXiv:1101.4448v1 [math.NA], to appear in Proc. Amer. Math. Soc.] for $\\beta = 1$. This is in analogy to the generalization of the r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5157","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"}