{"paper":{"title":"Elementary numerical methods for double integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Cameron Grant, Erik Talvila","submitted_at":"2019-05-14T19:25:08Z","abstract_excerpt":"Approximations to the integral $\\int_a^b\\int_c^d f(x,y)\\,dy\\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\\leq p\\leq\\infty$. We assume ${\\lVert f_{xy}\\rVert}_p$ is bounded (integration over $[a,b]\\times[c,d]$), assume ${\\lVert f_x(\\cdot,c)\\rVert}_p$ and ${\\lVert f_x(\\cdot,d)\\rVert}_p$ are bounded (integration over $[a,b]$), and assume ${\\lVert f_y(a,\\cdot)\\rVert}_p$ and ${\\lVert f_y(b,\\cdot)\\rVert}_p$ are bounded (integration over $[c,d]$). The methods are elementary, using only integration by parts and H\\\"older's inequal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.05805","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"}