Uniform approximation by polynomials with integer coefficients

• Autorzy: Lipnicki A.

Identyfikatory

DOI

10.7494/OpMath.2016.36.4.489

• Języki publikacji: EN

Abstrakty

EN

Let r, n be positive integers with n ≥ 6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] are integers for k = 0,…, r — 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.

Słowa kluczowe

EN

approximation by polynomials with integer coefficients; lattice; covering radius

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2016
• Tom: Vol. 36, no. 4
• Strony: 489--498
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Lipnicki A.

• University of Łódź Faculty of Mathematics and Computer Science ul. Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [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 Lp(0,l), 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.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.