Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

• Języki publikacji: EN

Abstrakty

EN

For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form
$P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ L(∑_{j=1}^n |a_j|^p)^{1/p}$, $a_j ∈ ℂ$,
such that $(x-1)^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let $μ_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that
$|Q(0)| > 1/L (∑_{j=1}^n |Q(j)|^q)^{1/q}$.
We find the size of $κ_p(n,L)$ and $μ_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about $μ_∞(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.

Kategorie tematyczne

• 26C10: Polynomials: location of zeros
• 11C08: Polynomials
• 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 172
• Numer: 3
• Strony: 271-284

Daty

wydano

2016

Twórcy

Identyfikatory

DOI

10.4064/aa8129-11-2015