Best-possible one-sided approximations in the classes $W_\alpha^rV$ ($r>-1$) by trigonometric polynomials in the metric of $L_1$

The quantities $\sup\limits_{f\in W_\alpha^rV}\Hat{\Hat E}_n(f)_1$ ($r>-1$, $-\infty<\alpha<\infty$, $n=1,2\dots)$ are calculated, where $\Hat{\Hat E}_n(f)_1$ is the best approximation from above of the function $f$ by trigonometric polynomials of order $\le n-1$ in the metric of $L_1$.