{"paper":{"title":"Approximation by polynomials in Sobolev spaces with Jacobi weight","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.CA","authors_text":"Yuan Xu","submitted_at":"2016-08-14T17:06:25Z","abstract_excerpt":"Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\\alpha,\\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\\alpha,\\beta}$. This requires simultaneous approximation of a function and its consecutive derivatives up to $s$-th order with $s \\le r$. We provide sharp error estimates given in terms of $E_n(f^{(r)})_{L^p(w_{\\alpha,\\beta})}$, the error of best approximation to $f^{(r)}$ by polynomials in $L^p(w_{\\alpha,\\beta})$, and an explicit construction of the polynomials that approximate simultaneously with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04114","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"}