On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$

Wojciech Banaszczyk, and Artur Lipnicki. "On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$." Annales Polonici Mathematici 115.2 (2015): 123-144. <http://eudml.org/doc/280296>.

@article{WojciechBanaszczyk2015,
abstract = {The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_\{n,r\}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r(1-x)^r$, r ≥ 0, and let $P_\{n,r\}^ℤ ⊂ P_\{n,r\}$ be the set of polynomials with integer coefficients. Let $μ(P_\{n,r\}^ℤ;L_p)$ be the maximal distance of elements of $P_\{n,r\}$ from $P_\{n,r\}^ℤ$ in $L_p(0,1)$. We give rather precise quantitative estimates of $μ(P_\{n,r\}^ℤ;L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ(P_\{n,r\}^ℤ;L_p)$ for p ≠ 2. It follows that $μ(P_\{n,r\}^ℤ;L_p) ≍ n^\{-2r-2/p\}$ as n → ∞. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].},
author = {Wojciech Banaszczyk, Artur Lipnicki},
journal = {Annales Polonici Mathematici},
keywords = {polynomials with integer coefficients; polynomial approximation; lattice; covering radius},
language = {eng},
number = {2},
pages = {123-144},
title = {On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$},
url = {http://eudml.org/doc/280296},
volume = {115},
year = {2015},
}

TY - JOUR
AU - Wojciech Banaszczyk
AU - Artur Lipnicki
TI - On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$
JO - Annales Polonici Mathematici
PY - 2015
VL - 115
IS - 2
SP - 123
EP - 144
AB - The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r(1-x)^r$, r ≥ 0, and let $P_{n,r}^ℤ ⊂ P_{n,r}$ be the set of polynomials with integer coefficients. Let $μ(P_{n,r}^ℤ;L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^ℤ$ in $L_p(0,1)$. We give rather precise quantitative estimates of $μ(P_{n,r}^ℤ;L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ(P_{n,r}^ℤ;L_p)$ for p ≠ 2. It follows that $μ(P_{n,r}^ℤ;L_p) ≍ n^{-2r-2/p}$ as n → ∞. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].
LA - eng
KW - polynomials with integer coefficients; polynomial approximation; lattice; covering radius
UR - http://eudml.org/doc/280296
ER -