Uniform approximation to \({}X{}^ \beta\) by Sinc functions (Q1176333)

It is well known that the error of the best uniform approximation to \(| x|^ \beta\), \(\beta>0\), on \([-1,1]\) decays like \(O(e^{- c\sqrt n})\), as \(n\to\infty\), where \(n\) is the dimension of the space of the approximating functions. In this paper the author proves that \[ E_ n(| x|^ \beta)\leq{M\over 4\pi}e^{-\pi n/\log n} \] uniformly for \(0<\beta_ 0\leq \beta<\beta_ 1\), where \(E_ n(| x|^ \beta)\) is the error of the uniform best approximation to \(| x|^ \beta\) on \([-1,1]\) by linear combinations of the basis functions \(B_{n,k}=S(k,h)\circ \sin h^{-1}(\cos h^{-1}({1\over | x|}))\), \(h=\log n/n\), \(k=-n,\ldots,n\) and \[ S(k,h)(x)={\sin[\pi/h(x-kh)]\over \pi/h(x-kh)}. \]