Free-knot Splines Approximation of s-monotone Functions

Abstract. Let I be a finite interval and r, s ∈ N. Given a set M , of functions defined on I, denote by ∆+M the subset of all functions y ∈ M such that the s-difference ∆τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆+W r p , the class of functions x on I with the seminorm ‖x‖Lp ≤ 1, such that ∆τ x ≥ 0, τ > 0. Let Mn(hk) := Pn i=1 cihk(wit − θi) | ci, wi, θi ∈ R , be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions hk(t) = t k +, t ∈ R, k ∈ N0. We give two-sided estimates both of the best unconstrained approximation E ∆+W r p , Mn(hk) Lq , k = r−1, r, s = 0, 1, . . . , r+1,