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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 $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.08643","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-28T22:51:03Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"33f21bcc0d277ea5fcbaa145661cb08eea0e86e1a132b6e420e536d5c5299833","abstract_canon_sha256":"913237e579218a53a85ac4a019450899c15a2350c85a0dee1619e514ef734183"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:03.247479Z","signature_b64":"B/42Q1WZce2WSBqajS85e9fn2GPedpmVyzEoZOkWWa/Kcs51xVZsBvYrxpU5mdcYoo2se8V6vzJ9ke5D01wKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20ab4e7f789ac40c0cdecf4d02f6160abaacfddfb79109eb730e2a9cea17e5c4","last_reissued_at":"2026-05-18T01:16:03.246764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:03.246764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\int_a^bf(x)\\,dx$. 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