Approximating monomials using Chebyshev polynomials

Abstract

This paper considers the approximation of a monomial $x^n$ over the interval $[-1,1]$ by a lower-degree polynomial. This polynomial approximation can be easily computed analytically and is obtained by truncating the analytical Chebyshev series expansion of $x^n$. The error in the polynomial approximation in the supremum norm has an exact expression with an interesting probabilistic interpretation. We use this interpretation along with concentration inequalities to develop a useful upper bound for the error.