On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L ₁-approximations of classes W ^r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes W ^r H ^ω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ?, k ∈ ?, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.