Best uniform rational approximation of $x^{\alpha }$ on $[0,1]$

Abstract

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

7 pages