{"paper":{"title":"Marcinkiewicz--Zygmund measures on manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"F. Filbir, H. N. Mhaskar","submitted_at":"2010-06-26T09:00:12Z","abstract_excerpt":"Let ${\\mathbb X}$ be a compact, connected, Riemannian manifold (without boundary), $\\rho$ be the geodesic distance on ${\\mathbb X}$, $\\mu$ be a probability measure on ${\\mathbb X}$, and $\\{\\phi_k\\}$ be an orthonormal system of continuous functions, $\\phi_0(x)=1$ for all $x\\in{\\mathbb X}$, $\\{\\ell_k\\}_{k=0}^\\infty$ be an nondecreasing sequence of real numbers with $\\ell_0=1$, $\\ell_k\\uparrow\\infty$ as $k\\to\\infty$, $\\Pi_L:={\\mathsf {span}}\\{\\phi_j : \\ell_j\\le L\\}$, $L\\ge 0$. We describe conditions to ensure an equivalence between the $L^p$ norms of elements of $\\Pi_L$ with their suitably discre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5123","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"}