On Certain Extremal Problems for Polynomials

For a given set of n natural numbers v1, ..., vn there exists a unique monic polynomial τ∗(x) ≔ Πnj=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 TN(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 Πnj=1 (x − xj)vj, where −1 ≤ x1 ≤ ••• ≤ xn ≤ 1. Here, for polynomials of the form P(z) ≔ c Πnj=1 (z − xj)vj, −1 ≤ x1 ≤ ••• ≤ xn ≤ 1, where v1,..., vn, are prescribed we consider the problems of estimating (i) the Lp 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.