{"paper":{"title":"Optimal Stencils in Sobolev Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Oleg Davydov, Robert Schaback","submitted_at":"2016-11-15T09:10:09Z","abstract_excerpt":"This paper proves that the approximation of pointwise derivatives of order $s$ of functions in Sobolev space $W_2^m(\\R^d)$ by linear combinations of function values cannot have a convergence rate better than $m-s-d/2$, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by {\\em scalable} approximations that are exact on polynomials of order at least $\\lfloor m-d/2\\rfloor +1$, proving that the rates are optimal for given $m,\\,s,$ and $d$. And, for a fixed node set $X\\subset\\R^d$, the convergence rate in any Sobolev space $W_2^m(\\Om"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04750","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":""},"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"}