Opuscula Math. 36, no. 4 (2016), 489-498
http://dx.doi.org/10.7494/OpMath.2016.36.4.489

 
Opuscula Mathematica

Uniform approximation by polynomials with integer coefficients

Artur Lipnicki

Abstract. Let \(r\), \(n\) be positive integers with \(n\ge 6r\). Let \(P\) be a polynomial of degree at most \(n\) on \([0,1]\) with real coefficients, such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\). It is proved that there is a polynomial \(Q\) of degree at most \(n\) with integer coefficients such that \(|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}\) for \(x\in[0,1]\), where \(C_1\), \(C_2\) are some numerical constants. The result is the best possible up to the constants.

Keywords: approximation by polynomials with integer coefficients, lattice, covering radius.

Mathematics Subject Classification: 41A10, 52C07.

Full text (pdf)

  1. E. Aparicio Bernardo, On some properties of polynomials with integral coefficients and on the approximation of functions in the mean by polynomials with integral coefficients, Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 303-318 [in Russian].
  2. W. Banaszczyk, A. Lipnicki On the lattice of polynomials with integer coefficients: the covering radius in \(L_p(0,1)\), Ann. Polon. Math. 115 (2015) 2, 123-144.
  3. L.B.O. Ferguson, Approximation by Polynomials with Integer Coefficients, Amer. Math. Society, Providence, R.I., 1980.
  4. L.B.O. Ferguson, What can be approximated by polynomials with integer coefficients, Amer. Math. Monthly 113 (2006), 403-414.
  5. L.V. Kantorowicz, Neskol'ko zamecanii o priblizenii k funkciyam posredstvom polinomov celymi koefficientami, Izvestiya Akademii Nauk SSSR Ser. Mat. (1931), 1163-1168.
  6. R.M. Trigub, Approximation of functions by polynomials with integer coefficients, Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izw.] (1962), 261-280.
  • Artur Lipnicki
  • University of Łódź, Faculty of Mathematics and Computer Science, ul. Banacha 22, 90-238 Łódź, Poland
  • Communicated by Henryk Hudzik.
  • Received: 2015-12-04.
  • Revised: 2016-01-29.
  • Accepted: 2016-02-01.
  • Published online: 2016-04-01.
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Cite this article as:
Artur Lipnicki, Uniform approximation by polynomials with integer coefficients, Opuscula Math. 36, no. 4 (2016), 489-498, http://dx.doi.org/10.7494/OpMath.2016.36.4.489

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