{"paper":{"title":"Approximation of mixed order Sobolev functions on the $d$-torus -- Asymptotics, preasymptotics and $d$-dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Thomas Kuehn, Tino Ullrich, Winfried Sickel","submitted_at":"2013-12-22T14:20:57Z","abstract_excerpt":"We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other words, we study the approximation numbers of the Sobolev embeddings $H^s_{\\rm mix}(\\mathbb{T}^d)\\hookrightarrow L_2(\\mathbb{T}^d)$, with particular emphasis on the dependence on the dimension $d$. For any fixed smoothness $s>0$, we find the exact asymptotic behavior of the constants as $d\\to\\infty$. We observe super-exponential decay of the constants in $d$, if $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6386","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"}