Free-knot splines approximation of s-monotone functions

Let I be a finite interval and r, s ∈ 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 Δ₊^sW_p^r, the class of functions x on I with the seminorm ∥x ^(r)∥_Lp ≤ 1, such that Δ_τ ^sx ≥ 0, τ > 0. Let M_n(h_k):= {∑_i=1ⁿc_ih_k(w_it- θ_i) | c_i, w_i, θ_i ∈ ℝ}, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h_k(t) = t₊ ^k, t ∈ ℝ, k ∈ ℕ₀. We give two-sided estimates both of the best unconstrained approximation E(Δ ₊^sW_p^r,M_n(h _k))_Lq, k = r - 1, r, s = 0, 1,...,r + 1, and of the best s-monotonicity preserving approximation E(Δ₊^sW _p^r,Δ₊^sM_n(h _k))_Lq, k = r - 1, r, s = 0, 1,...,r + 1. The most significant results are contained in theorem 2.2.