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

Abstract

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by $\Delta ^{s}_{ + }$ M the subset of all functions y ∈ M such that the s-difference $\Delta ^{s}_{\tau } y(\cdot)$ is nonnegative on I, ∀τ > 0. Further, denote by $W^{r}_{p}$ the Sobolev class of functions x on I with the seminorm $\|x^{(r)}\|_{L_p}\le 1$. Also denote by Σ _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 $E\bigl(\Delta^s_+W^r_p,\Sigma_{\nu,n}\bigr)_{L_q}$ and of the best s-monotonicity preserving approximation $E\bigl(\Delta^s_+W^r_p,\Delta^s_+\Sigma_{\nu,n}\bigr)_{L_q}$.