Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

For functions $f(x)\in KH^{(\alpha)}$ (satisfying the Lipschitz condition of order $\alpha$ ($0<\alpha<1$) with constant $K$ on $[-1, 1]$), the existence is proved of a sequence $P_n(f;\,x)$ of algebraic polynomials of degree $n=1,\,2,\,\dots$, such that $|f(x)-P_{n-1}(f;\,x)|\leqslant\sup\limits_{f\in KH^{(\alpha)}}E_n(f)[(1-x^2)^{\alpha/2}+o(1)]$ when $n\to\infty$, uniformly for $x\in[-1,\,1]$ , where $E_n(f)$ is the best approximation of $f(x)$ by polynomials of degree not higher than $n$.