Exact values of best approximations for classes of periodic functions by splines of deficiency 2

Abstract

We obtain exact values of best L1-approximations for the classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F as well as for the classes WrHω of periodic functions whose rth derivative has a given convex (up) majorant ω(t) of the modulus of continuity by subspaces of polynomial splines of order m ≥ r + 1 of deficiency 2 with nodes at the points 2kπ/n, n ∈ ℕ, k ∈ ℤ. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding function classes.