{
  "id": "1612.08020",
  "version": "v1",
  "published": "2016-12-23T16:03:40.000Z",
  "updated": "2016-12-23T16:03:40.000Z",
  "title": "Best approximations and moduli of smoothness of functions and their derivatives in $L_p$, $0<p<1$",
  "authors": [
    "Yurii Kolomoitsev"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\\,r\\in \\mathbb{N}$ one has $ \\omega_{r+k}(f,\\d)_p\\leq C({p,k,r})\\d^{r+\\frac{1}{p}-1}\\(\\int_0^\\d\\frac{\\omega_{k}(f^{(r)},t)_p^p}{t^{2-p}}{\\rm d}t\\)^\\frac{1}{p},$ where the function $f$ is such that $f^{(r-1)}$ is absolutely continuous. Similar inequalities are obtained for the Ditzian-Totik moduli of smoothness and the error of the best approximation of functions by trigonometric and algebraic polynomials and splines. As an application, positive results about simultaneous approximation of a function and its derivatives by the mentioned approximation methods in the spaces $L_p$, $0<p<1$, are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-23T16:03:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "best approximation",
      "smoothness",
      "derivatives",
      "mentioned approximation methods",
      "similar inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}