{"paper":{"title":"Optimal Quadrature Formulas for the Sobolev Space $H^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Erich Novak, Shun Zhang","submitted_at":"2016-09-05T13:28:26Z","abstract_excerpt":"We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space $H^1([0,1])$. We obtain general formulas for the worst case error depending on the nodes $x_j$. A particular case is the computation of Fourier coefficients, where the oscillatory weight is given by $\\rho_k(x) = \\exp(- 2 \\pi i k x)$. Here we study the question whether equidistant nodes are optimal or not. We prove that this depends on $n$ and $k$: equidistant nodes are optimal if $n \\ge 2.7 |k| +1 $ but might be suboptimal for small $n$. In particular, the equidistant nodes $x_j = j/ |k|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01146","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"}