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

• Autorzy: Wojciech Banaszczyk , Artur Lipnicki
• Języki publikacji: EN

Abstrakty

EN

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)].

Kategorie tematyczne

• 52C07: Lattices and convex bodies in n dimensions
• 41A10: Approximation by polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2015
• Tom: 115
• Numer: 2
• Strony: 123-144

Daty

wydano

2015

Twórcy

autor

Wojciech Banaszczyk

• Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Łódź, Poland

autor

Artur Lipnicki

• Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Łódź, Poland

Identyfikatory

DOI

10.4064/ap115-2-2