Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofxk by polynomials of degreen (n<k) on the interval [−1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann2 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 numberpk,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.