{"paper":{"title":"Nonlinear Piecewise Polynomial Approximation and Multivariate $BV$ spaces of a Wiener--L.~Young Type. I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Yu. Brudnyi","submitted_at":"2015-11-12T17:06:33Z","abstract_excerpt":"The named space denoted by $V_{pq}^k$ consists of $L_q$ functions on $[0,1)^d$ of bounded $p$-variation of order $k\\in\\mathbb N$. It generalizes the classical spaces $V_p(0,1)$ ($=V_{p\\infty}^1$) and $BV([0,1)^d)$ ($V_{1q}^1$ where $q:=\\frac d{d-1}$) and closely relates to several important smoothness spaces, e.g., to Sobolev spaces over $L_p$, $BV$ and $BMO$ and to Besov spaces.\n  The main approximation result concerns the space $V_{pq}^k$ of \\textit{smoothness} $s:=d\\left(\\frac1p-\\frac1q\\right)\\in(0,k]$. It asserts the following:\n  Let $f\\in V_{pq}^k$ are of smoothness $s\\in(0,k]$ and $N\\in\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03971","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"}