Geodesic
Best-possible one-sided approximations in the classes $W_\alpha^rV$ ($r>-1$) by trigonometric polynomials in the metric of $L_1$
Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 357-370

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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 $L_1$. 
@article{MZM_1977_22_3_a4,
     author = {V. G. Doronin and A. A. Ligun},
     title = {Best-possible one-sided approximations in the classes $W_\alpha^rV$ ($r>-1$) by trigonometric polynomials in the metric of $L_1$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {357--370},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a4/}
}
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V. G. Doronin; A. A. Ligun. Best-possible one-sided approximations in the classes $W_\alpha^rV$ ($r>-1$) by trigonometric polynomials in the metric of $L_1$. Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 357-370. http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a4/