Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail.