{
  "id": "1809.07733",
  "version": "v1",
  "published": "2018-09-20T16:50:46.000Z",
  "updated": "2018-09-20T16:50:46.000Z",
  "title": "Reverse Markov- and Bernstein-type inequalities for incomplete polynomials",
  "authors": [
    "Tamás Erdélyi"
  ],
  "comment": "submitted to Journal of Approximation Theory",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let ${\\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$\\|f\\|_A := \\sup_{x \\in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A \\subset {\\Bbb R}$. Let $$V_a^b(f) := \\int_a^b{|f^{\\prime}(x)| \\, dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \\frac nk\\leq \\min_{P \\in {\\mathcal P}_{n,k}}{\\frac{\\|P^{\\prime}\\|_{[0,1]}}{V_0^1(P)}} \\leq \\min_{P \\in {\\mathcal P}_{n,k}}{\\frac{\\|P^{\\prime}\\|_{[0,1]}}{|P(1)|}} \\leq c_2 \\left( \\frac nk + 1 \\right)$$ for all integers $n \\geq 1$ and $k \\geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \\left(\\frac nk\\right)^{1/2} \\leq \\min_{P \\in {\\mathcal P}_{n,k}}{\\frac{\\|P^{\\prime}(x)\\sqrt{1-x^2}\\|_{[0,1]}}{V_0^1(P)}} \\leq \\min_{P \\in {\\mathcal P}_{n,k}}{\\frac{\\|P^{\\prime}(x)\\sqrt{1-x^2}\\|_{[0,1]}}{|P(1)|}} \\leq c_2 \\left(\\frac nk + 1\\right)^{1/2}$$ for all integers $n \\geq 1$ and $k \\geq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-20T16:50:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A17"
    ],
    "keywords": [
      "bernstein-type inequalities",
      "incomplete polynomials",
      "algebraic polynomials",
      "absolute constants",
      "real coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}