{"paper":{"title":"Adaptive approximation by optimal weighted least squares methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Giovanni Migliorati","submitted_at":"2018-07-01T21:51:46Z","abstract_excerpt":"Given any domain $X\\subseteq \\mathbb{R}^d$ and a probability measure $\\rho$ on $X$, we study the problem of approximating in $L^2(X,\\rho)$ a given function $u:X\\to\\mathbb{R}$, using its noiseless pointwise evaluations at random samples. For any given linear space $V\\subset L^2(X,\\rho)$ with dimension $n$, previous works have shown that stable and optimally converging Weighted Least-Squares (WLS) estimators can be constructed using $m$ random samples distributed according to an auxiliary probability measure $\\mu$ that depends on $V$, with $m$ being linearly proportional to $n$ up to a logarithm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00402","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"}