{
  "id": "1511.03971",
  "version": "v1",
  "published": "2015-11-12T17:06:33.000Z",
  "updated": "2015-11-12T17:06:33.000Z",
  "title": "Nonlinear Piecewise Polynomial Approximation and Multivariate $BV$ spaces of a Wiener--L.~Young Type. I",
  "authors": [
    "Yu. Brudnyi"
  ],
  "comment": "37 pages",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "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. 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: Let $f\\in V_{pq}^k$ are of smoothness $s\\in(0,k]$ and $N\\in\\mathbb N$. There exist a family $\\Delta_N$ of $N$ dyadic subcubes of $[0,1)^d$ and a piecewise polynomial $g_N$ over $\\Delta_N$ of degree $k-1$ such that \\[ \\|f-g_N\\|_q\\leqslant CN^{-s/d}|f|_{V_{pq}^k}. \\] This implies the similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman--Solomyak paper \\cite{BS} problem of approximation of functions from $W_p^k([0,1)^d)$ in $L_q([0,1)^d)$ when ever $\\frac kd=\\frac1p-\\frac1q$ and $q<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-12T17:06:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A46",
      "41A15"
    ],
    "keywords": [
      "nonlinear piecewise polynomial approximation",
      "main approximation result concerns",
      "multivariate",
      "important smoothness spaces",
      "besov spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151103971B"
    }
  }
}