Approximating monomials using Chebyshev polynomials

TL;DR

This paper explores how Chebyshev polynomial truncation can efficiently approximate monomials over [-1,1], providing exact error formulas and probabilistic bounds to improve understanding of approximation accuracy.

Contribution

It introduces a method to approximate monomials with truncated Chebyshev series, deriving exact error expressions and probabilistic bounds for the approximation error.

Findings

Exact error expression for Chebyshev polynomial approximation

Probabilistic interpretation of approximation error

Upper bounds for approximation error using concentration inequalities

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.