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.