TY  - JOUR
ID  - Lipnicki2016
LB  - Lipnicki2016
AU  - Lipnicki, Artur
TI  - Uniform approximation by polynomials with integer coefficients
ST  - Uniform approximation by polynomials with integer coefficients
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2016
VL  - 36
IS  - 4
SP  - 489
EP  - 498
DO  - http://dx.doi.org/10.7494/OpMath.2016.36.4.489
UR  - http://dx.doi.org/10.7494/OpMath.2016.36.4.489
AB  - 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.
KW  - approximation by polynomials with integer coefficients
KW  - lattice
KW  - covering radius
SN  - 1232-9274
ER  -