Piecewise polynomial approximation

For best piecewise polynomial approximation $\mathscr{E}_n=\mathscr{E}_n(f;[0, 1])$ of a function $f$, which is continuous on the interval $[0,1]$ and admits a bounded analytic continuation onto the disk $K=\{z: |z-1|<1\}$, the relation $\mathscr{E}_n=O[\omega_f(e^{-\sqrt{n}})]$ is valid.