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.

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• Artur Lipnicki
• University of Łódź, Faculty of Mathematics and Computer Science, ul. Banacha 22, 90-238 Łódź, Poland
• Communicated by Henryk Hudzik.
• Revised: 2016-01-29.
• Accepted: 2016-02-01.
• Published online: 2016-04-01.