{"paper":{"title":"Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps","license":"","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.CA","authors_text":"Feng Dai, Heping Wang","submitted_at":"2007-03-26T18:44:05Z","abstract_excerpt":"Let $\\Pi_n^d$ denote the space of all spherical polynomials of degree at most $n$ on the unit sphere $\\sph$ of $\\mathbb{R}^{d+1}$, and let $d(x, y)$ denote the usual geodesic distance $\\arccos x\\cdot y$ between $x, y\\in \\sph$. Given a spherical cap $$ B(e,\\al)=\\{x\\in\\sph: d(x, e) \\leq \\al\\}, (e\\in\\sph, \\text{$\\al\\in (0,\\pi)$ is bounded away from $\\pi$}),$$ we define the metric $$\\rho(x,y):=\\frac 1{\\al} \\sqrt{(d(x, y))^2+\\al(\\sqrt{\\al-d(x, e)}-\\sqrt{\\al-d(y,e)})^2},\n  $$ where $x, y\\in B(e,\\al)$. It is shown that given any $\\be\\ge 1$, $1\\leq p<\\infty$ and any finite subset $\\Ld$ of $B(e,\\al)$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703768","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0703768/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}