Exact Values of the Best (𝛼, β)-Approximations for the Classes of Convolutions with Kernels that Do Not Increase the Number of Sign Changes

We obtain the exact values of the best (𝛼, β) -approximations for the classes K ∗ F of periodic functions K ∗ f such that f belongs to a given rearrangement-invariant set F and K is 2𝜋 -periodic kernel that does not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.