{"paper":{"title":"Uniform recovery of high-dimensional $C^r$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"David Krieg","submitted_at":"2018-05-16T10:03:31Z","abstract_excerpt":"We consider functions on the $d$-dimensional unit cube whose partial derivatives up to order $r$ are bounded by one. It is known that the minimal number of function values that is needed to approximate the integral of such functions up to the error $\\varepsilon$ is of order $(d/ \\varepsilon)^{d/r}$. Among other things, we show that the minimal number of function values that is needed to approximate such functions in the uniform norm is of order $(d^{r/2} /\\varepsilon)^{d/r}$ whenever $r$ is even."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06220","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"}