Approximation of monomials by lower degree polynomials

Abstract

Our topic is the uniform approximation of xk by polynomials of degree n (n<k) on the interval [-1, 1]. Our major result indicates that good approximation is possible when k is much smaller than n2 and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity, pk,n, which can be identified with a certain probability. The number pk,n is in fact the probability that when a (fair) coin is tossed k times the magnitude of the difference between the number of heads and the number of tails exceeds n. © 1976 Birkhäuser Verlag.