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 Δ ^s₊ M the subset of all functions y∈M such that the s-difference Δ ^s_τ y(⋅) is nonnegative on I, ∀τ>0. Further, denote by Δ ^s₊ W ^r _p , the class of functions x on I with the seminorm ‖x ^(r)‖L _p ≤1, such that Δ ^s_τ x≥0, τ>0. Let M _n (h _k ):={∑ _i=1 ⁿ c _i h _k (w _i t−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∈N ₀. We give two-sided estimates both of the best unconstrained approximation E(Δ ^s₊ W ^r _p ,M _n (h _k ))L _q , k=r−1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E(Δ ^s₊ W ^r _p ,Δ ^s₊ M _n (h _k ))L _q , k=r−1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.