Free-knot Splines Approximation of s-monotone Functions

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by ₊^sM the subset of all functions yM such that the s-difference ^sy() is nonnegative on I, >0. Further, denote by ₊^sW_p^r, the class of functions x on I with the seminorm x^(r)L_p1, such that ^sx0, >0. Let M_n(h_k):={_i=1ⁿc_ih_k(w_it–_i)c_i,w_i, _iR, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h_k(t)=t₊^k, tR, kN₀. We give two-sided estimates both of the best unconstrained approximation E(₊^sW_p^r,M_n(h_k))L_q, k=r–1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E(₊^sW_p^r,₊^sM_n(h_k))L_q, k=r–1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.