We obtained the exact values of the best $L_1$-approximations of the classes $K*F$ ($r\in \mathbb{N}$) of periodic functions $K*f$ such that $f$ belongs to a given rearrangement-invariant set $F$ and $K$ is $2\pi$-periodic, not increasing oscillation, kernel, by subspaces of generalized polynomial splines with nodes at points $2k\pi / n$ ($n\in \mathbb{N}$, $k\in \mathbb{Z}$). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.