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

Uniform approximation by polynomials with integer coefficients


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.
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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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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