On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$
Tom 115 / 2015
Streszczenie
The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, $1\le p\le \infty $. Let $\boldsymbol {P}_{n,r}$ be the space of polynomials of degree $\le n$ which are divisible by the polynomial $x^r(1-x)^r$, $r\ge 0$, and let $\boldsymbol {P}_{n,r}^\mathbb {Z}\subset \boldsymbol {P}_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)$ be the maximal distance of elements of $\boldsymbol {P}_{n,r}$ from $\boldsymbol {P}_{n,r}^\mathbb {Z}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_2)$ for $n\gtrsim 6r$. Then we obtain similar, somewhat less precise, estimates of $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)$ for $p\not =2$. It follows that $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)\asymp n^{-2r-2/p}$ as $n\to \infty $. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].
