{"paper":{"title":"Best polynomial approximation on the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Miguel Pinar, Yuan Xu","submitted_at":"2016-09-18T17:09:11Z","abstract_excerpt":"Let $E_n(f)_\\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\\varpi_\\mu, \\mathbb{B}^d)$, where $\\mathbb{B}^d$ is the unit ball in $\\mathbb{R}^d$ and $\\varpi_\\mu(x) = (1-\\|x\\|^2)^\\mu$ for $\\mu > -1$. Our main result shows that, for $s \\in \\mathbb{N}$, $$ E_n(f)_\\mu \\le c n^{-2s}[E_{n-2s}(\\Delta^s f)_{\\mu+2s} + E_{n}(\\Delta_0^s f)_{\\mu}], $$ where $\\Delta$ and $\\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05515","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"}