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 ₊ ^s M the subset of all functions yM such that the s-difference ^s y() is nonnegative on I, >0. Further, denote by ₊ ^s W _p ^r , the class of functions x on I with the seminorm x ^(r)L _p 1, such that ^s x0, >0. Let M _n (h _k ):={ _i=1 ⁿ c _i h _k (w _i t– _i )c _i ,w _i , _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 , tR, kN ₀. We give two-sided estimates both of the best unconstrained approximation E( ₊ ^s W _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( ₊ ^s W _p ^r , ₊ ^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.