Integral inequalities for algebraic polynomials

Abstract

In this paper we consider two extremal problems for algebraic polynomials in L ² metrics.

• (1) Let P _n be the class of all algebraic polynomials $P(x) = \sum\nolimits_{{k = 0}}^n {{a_k}{x^k}}$ of degree at most n and ∥P∥ _dσ =(∫_ℝ∣P(x)∣² dσ(x))^1/2, where dσ(x) is a nonnegative measure on ℝ. We determine the best constant in the inequality ∣a _k ∣ ≤ Cn,k (dσ) ∥P∥dσ for k=0,1,…, n, when P ∈P _n and such that P(ξk) = 0, k = 1,…, m. The cases C _n,n (dσ) and C _n,n-1(dσ) were studied by Milovanović and Guessab [5], and only for the Legendre measure by Tariq [9].

• (2) Let $\hat{\mathcal{P}}$ _n be the set of all monic algebraic polynomials of degree N and ε _s be Mth roots of unity, i.e., ε _s = exp(i2πs/M), s = 0,1,…, M - 1. Polynomials orthogonal on the radial rays in the complex plane with respect to the inner product

  $$ \left( {f,g} \right) = \int_o^a {\left( {\sum\limits_{{s = 0}}^{{M - 1}} {f\left( {x{\varepsilon_s}} \right)\overline {g\left( {x{\varepsilon_s}} \right)} } } \right)} w(x)dx $$

  have been introduced and studied recently in [3]. Here, w is a weight function and 0 < a ≤ +∞. We consider the extremal problem

  $$ \mathop{{\inf }}\limits_{{P \in {{\hat{\mathcal{P}}}_N}}} \int_0^a {\left( {\sum\limits_{{s = 0}}^{{M - 1}} {{{\left| {P\left( {x{\varepsilon_s}} \right)} \right|}^2}} } \right)w(x)dx,} $$

  as well as some inequalities for coefficients of polynomials under some restrictions of the polynomial class.