Geodesic
Best approximation by splines on classes of periodic functions in the metric of $L$
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 655-664.

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We have obtained the exact value of the upper bound on the best approximations in the metric of $L$ on the classes $W^rH^\omega$ of functions $f\in C_{2\pi}^r$ for which $|f^{(r)}(x')-f^{(r)}(x'')|\le\omega(|x'-x''|)$ [$\omega(t)$ is the upwards-convex modulus of continuity] by subspaces of $r$-th order polynomial splines of defect 1 with respect to the partitioning $k\pi/n$. 
@article{MZM_1976_20_5_a3,
     author = {N. P. Korneichuk},
     title = {Best approximation by splines on classes of periodic functions in the metric of $L$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {655--664},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a3/}
}
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N. P. Korneichuk. Best approximation by splines on classes of periodic functions in the metric of $L$. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 655-664. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a3/