An Extremal Problem Concerning the Derivatives of a Polynomial at Its Roots

Abstract

Let w be a monic polynomial of degree n+1 with roots xj in the interval [−1, 1]. We consider the problem of finding the roots xj for which the minimum of ¦w′(xj)¦, for 0≤j≤n, is as large as possible. We prove that the Clenshaw–Curtis points cos(jπ/n) are the only solution when n is even and that they get asymptotically close to the solution for odd values of n, as n goes to infinity. Our problem is related to the problem of minimizing the norm of inverse Vandermonde matrices.