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

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Banaszczyk W., Lipnicki A.

Annales Polonici Mathematici

2015

tom 115

nr 2

123-144

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)].

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Banaszczyk W.

1 Lipnicki A.

1 2015

Wyniki wyszukiwania

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