{"paper":{"title":"B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dinh D\\~ung","submitted_at":"2010-09-22T15:29:00Z","abstract_excerpt":"Let $\\xi = \\{x^j\\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube ${\\II}^d:=[0,1]^d$, and $\\Phi = \\{\\phi_j\\}_{j =1}^n$ a family of $n$ functions on ${\\II}^d$. We define the linear sampling algorithm $L_n(\\Phi,\\xi,\\cdot)$ for an approximate recovery of a continuous function $f$ on ${\\II}^d$ from the sampled values $f(x^1), ..., f(x^n)$, by $$L_n(\\Phi,\\xi,f)\\ := \\ \\sum_{j=1}^n f(x^j)\\phi_j$$.\n  For the Besov class $B^\\alpha_{p,\\theta}$ of mixed smoothness $\\alpha$ (defined as the unit ball of the Besov space $\\MB$), to study optimality of $L_n(\\Phi,\\xi,\\cdot)$ in $L_q({\\II}^d)$ we use the quan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4389","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"}