{
  "id": "1401.2055",
  "version": "v1",
  "published": "2014-01-09T16:19:35.000Z",
  "updated": "2014-01-09T16:19:35.000Z",
  "title": "Approximation by Genuine $q$-Bernstein-Durrmeyer Polynomials in Compact Disks in the case $q > 1$",
  "authors": [
    "Nazim I. Mahmudov"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, whcih are no longer positive linear operators on $C[0,1]$. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in $\\left\\{ z\\in\\mathbb{C}:\\left\\vert z\\right\\vert <R\\right\\} $, $R>q,$ the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for $q>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-09T16:19:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A35",
      "30E10",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "compact disks",
      "approximation",
      "longer positive linear operators",
      "voronovskaja type theorem",
      "classical genuine bernstein-durrmeyer polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.2055M"
    }
  }
}