Best approximation by splines on classes of periodic functions in the metric of $L$

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 $k\pi/n$.