{"paper":{"title":"Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.CA","authors_text":"Erik Talvila","submitted_at":"2016-04-28T22:51:03Z","abstract_excerpt":"If $f\\!:\\![a,b]\\to\\R$ such that $f^{(n)}$ is integrable then integration by parts gives the formula \\begin{align*} &\\intab f(x)\\,dx = &\\frac{(-1)^n}{n!}\\sum_{k=0}^{n-1}(-1)^{n-k-1}\\left[ \\phi_n^{(n-k-1)}(a)f^{(k)}(a)- \\phi_n^{(n-k-1)}(b)f^{(k)}(b)\\right] +E_n(f), \\end{align*} where $\\phi_n$ is a monic polynomial of degree $n$ and the error is given by $E_n(f)=\\frac{(-1)^n}{n!}\\int_a^b f^{(n)}(x)\\phi_n(x)\\,dx$. This then gives a quadrature formula for $\\int_a^bf(x)\\,dx$. The polynomial $\\phi_n$ is chosen to optimize the error estimate under the assumption that $f^{(n)}\\in L^p([a,b])$ for some $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08643","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"}