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

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