Freeknot splines approximation of Sobolev-type classes of s-monotone functions

Let I be a finite interval, s ∈₀, and r,ν,n ∈. Given a set M, of functions defined on I, denote by Δ₊ ^sM the subset of all functions y M such that the s-difference Δ_τ^sy(̇) is nonnegative on I, ∀_τ > 0. Further, denote by W_p^r the Sobolev class of functions x on I with the seminorm ||x ^(r)||L_p ≤ 1. Also denote by ∑ _ν,n, the manifold of all piecewise polynomials of order ν and with n - 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation E(Δ₊^s W_p ^r, Σ_νn)_Lq and of the best s-monotonicity preserving approximation E(Δ₊^s W_p ^r, Σ_νn)_Lq.