On Certain Extremal Problems for Polynomials

For a given set of n natural numbers v 1, ..., v n there exists a unique monic polynomial τ∗( x) ≔ Π n j=1 ( x − x* j ) vj , where −1 < x* 1 < ··· < x* n < 1 along with n + 1 points −1 =: t* 0 < t* 1 < ··· < t* n ≔ 1 such that |τ∗( t* k )| = max − 1 ≤ x ≤ 1 |τ∗( x)\ for k = 0, ..., n. The polynomial τ∗ is a generalization of the Chebyshev polynomial T N ( x) ≔ cos( N are cos x) in the sense that its supremum norm on [−1, 1] is the smallest amongst all polynomials of the form Π n j=1 ( x − x j ) vj , where −1 ≤ x 1 ≤ ··· ≤ x n ≤ 1. Here, for polynomials of the form P( z) ≔ c Π n j=1 ( z − x j ) vj , −1 ≤ x 1 ≤ ··· ≤ x n ≤ 1, where v 1,..., v n, are prescribed we consider the problems of estimating (i) the L p norm of P ( k) on [−1, 1] for p ∈ [1, ∞], (ii) | P ( k) ( z)| at an arbitrary point outside the unit disk, given that the supremum norm of P on [−1, 1] does not exceed that of the (corresponding) generalized Chebyshev polynomial τ∗. It turns out that τ∗ is extremal for both the problems.