Best uniform rational approximation of $x^α$ on $[0,1]$

Abstract

A strong error estimate for the uniform rational approximation of $x^α$ on $[0,1]$ is given, and its proof is sketched. Let $E_{nn}(x^α,[0,1])$ denote the minimal approximation error in the uniform norm. Then it is shown that $$\lim_{n\to\infty}e^{2π\sqrt{αn}}E_{nn}(x^α,[0,1]) = 4^{1+α}|\sinπα|$$ holds true for each $α>0$.