{
  "id": "1602.07313",
  "version": "v1",
  "published": "2016-02-23T21:04:51.000Z",
  "updated": "2016-02-23T21:04:51.000Z",
  "title": "Yet another look at positive linear operators, $q$-monotonicity and applications",
  "authors": [
    "K. Kopotun",
    "D. Leviatan",
    "A. Prymak",
    "I. A. Shevchuk"
  ],
  "categories": [
    "math.CA",
    "math.FA",
    "math.NA"
  ],
  "abstract": "For each $q\\in{\\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\\leq k\\leq q$ and yield the estimate \\[ |f(x)-M_n(f, x)| \\leq c \\omega_2^{\\varphi^\\lambda} \\left(f, n^{-1} \\varphi^{1-\\lambda/2}(x) \\left(\\varphi(x) + 1/n \\right)^{-\\lambda/2} \\right) , \\] for $x\\in [0,1]$ and $\\lambda\\in [0, 2)$, where $\\varphi(x) := \\sqrt{x(1-x)}$ and $\\omega_2^{\\psi}$ is the second Ditzian-Totik modulus of smoothness corresponding to the \"step-weight function\" $\\psi$. In particular, this implies that the rate of best uniform $q$-monotone polynomial approximation can be estimated in terms of $\\omega_2^{\\varphi} \\left(f, 1/n \\right)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-23T21:04:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A10",
      "41A17",
      "41A25"
    ],
    "keywords": [
      "positive linear operators",
      "monotonicity",
      "applications",
      "construct positive linear polynomial approximation",
      "positive linear polynomial approximation operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}