Approximating monomials using Chebyshev polynomials

This paper considers the approximation of a monomial x over the interval [−1, 1] by a lowerdegree polynomial. This polynomial approximation can be easily computed analytically and is obtained by truncating the analytical Chebyshev series expansion of x. 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. Keywords— Chebyshev polynomials, Polynomial Approximation, Binomial Coefficients, Concentration inequalities 1 Motivation and Introduction We are interested in approximating the monomial x by a polynomial of degree 0 ≤ k < n over the interval [−1, 1]. The monomials 1, x, x, . . . form a basis for C[−1, 1], so it seems unlikely that we can represent a monomial in terms of lower degree polynomials. In Figure 1, we plot a few functions from the monomial basis over [0, 1]; the basis function look increasingly alike as we take higher and higher powers, i.e., they appear to “lose independence.” Numerical analysts often avoid the monomial basis in polynomial interpolation since they result in ill-conditioned Vandermonde matrices, leading to poor numerical performance in finite precision arithmetic. This loss of independence means that it is reasonable to approximate the monomial x as a linear combination of lower order monomials, i.e., a lower order polynomial approximation. The natural question to ask, therefore, is: how small can k be so that a well-chosen polynomial of degree k can accurately approximate x? 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x x x x x Figure 1: Visualization of a few monomials in the interval [0, 1]. ∗Department of Mathematics, North Carolina State University, Raleigh, NC. Email: asaibab@ncsu.edu.