On Extrapolation of Polynomials with Real Coefficients to the Complex Plane

Abstract

The problem of the greatest possible absolute value of the kth derivative of an algebraic polynomial of order n > k with real coefficients at a given point of the complex plane is considered. It is assumed that the polynomial is bounded by 1 on the interval [-1,1]. It is shown that the solution is attained for the polynomial κ · T_σ, where T_σ is one of the Zolotarev or Chebyshev polynomials and κ is a number.