{"paper":{"title":"Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Josef Dick","submitted_at":"2013-04-01T10:43:24Z","abstract_excerpt":"We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\\delta \\ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital $(t,\\alpha,s)$-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital $(t,\\alpha,s)$-s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0328","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"}