Some numerical results on best uniform polynomial approximation of X ^α on [0, 1]

Abstract

Let α be a positive number, and let E _n (x ^α; [0, 1]) denote the error of best uniform approximation to x ^α by polynomials of degree at most n on the interval [0, 1]. Russian mathematician S. N. Bernstein established the existence of a nonnegative constant β(α) such that β(α):=lim_n→∞(2n)^2α E _n (x ^α;[0, 1]) (α>0).

In addition, Bernstein showed that πβ(α)<Γ(2α)|sin(πα)| (α>0), and that Γ(2α)|sin(πα)|(1−1/(2α−1))<πβ(α) (α>1/2), so that the asymptotic behavior of β(α) is thus known when α → ∞.

Still, the problem of trying to determine β(α) more precisely, for all α>0, is intriguing. To this end, we have rigorously determined the numbers lcub;E _n (x ^α;[0, 1])rcub; ⁴⁰_n=1 for thirteen values of α, where these numbers were calculated with a precision of at least 200 significant digits. For each of these thirteen values of α. Richardson’s extrapolation was applied to the products lcub;(2n)^2α E _n (x ^α; [0, 1])rcub; ⁴⁰_n=1 to obtain estimates of β(α) to approximately 40 decimal places. Included are graphs of the points (α,β(α)) for the thirteen values of α that we considered.

Research was done while a National Science Foundation intern in parallel processing in the Mathematics and Computer Science Division, Argonne National Laboratory.

Research supported by the National Science Foundation.