{"paper":{"title":"Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Sebastian Mayer, Thomas K\\\"uhn, Tino Ullrich","submitted_at":"2015-05-04T13:41:05Z","abstract_excerpt":"In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\\|\\cdot\\|$ on $\\mathbb{R}^d$, and entropy numbers of the embedding $\\textrm{id}: \\ell_{\\|\\cdot\\|}^d \\to \\ell_\\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00631","kind":"arxiv","version":3},"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"}