Monotone approximation

Abstract

0. SHISHA How close can one approximate a monotone function by a monotone polynomial of degree ^ n, or a convex function by a convex polynomial of degree ^ nt This leads to the following general question. Let k and n be given, and suppose a real fuction / satisfies f {k) (x) ^ 0 throughout a closed, finite interval [a, &]. How close can one approximate / on [, b] by a polynomial of degree ^ n whose A th derivative, too, is ^ 0 there? We give an answer to the question.