On the best non-symmetric $L_1$-approximations by splines under constraints for their derivatives

We find exact values of non-symmetric $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s \in S_{2n,r-1}$ and $s \in S_{2n,r}$ ($S_{2n,r}$ is the set of $2\pi$-periodic polynomial splines of degree $r$, of defect 1, with knots in the points $k \pi / n$, $k \in \mathbb{Z}$) such that $\bigvee\limits_0^{2\pi} s^{(r-1)} \leqslant 1$ and $\| s^{(r)} \| \leqslant 1$ respectively when $r$ is even, and, as a corollary, we obtain exact values for the corresponding best one-side approximations.