{"paper":{"title":"Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Dinh D\\~ung, Tino Ullrich","submitted_at":"2013-11-07T03:14:32Z","abstract_excerpt":"We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^{\\alpha}_{p,\\theta}({\\mathbb G}^d)$ in the case $0 < p, \\theta \\le \\infty$ and $\\alpha > 1/p$, where ${\\mathbb G}^d$ is either the $d$-dimensional torus ${\\mathbb T}^d$ or the $d$-dimensional unit cube ${\\mathbb I}^d$. We prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from $B^{\\alpha}_{p,\\theta}({\\mathbb T}^2)$ in the case $1\\leq p \\leq \\infty$, $0 < \\theta \\leq \\infty$, $\\alpha>1/p$. A non-periodic modi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1563","kind":"arxiv","version":2},"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"}