Density Results by Deep Neural Network operators with Integer weights

In the present paper, a new family of multi-layers (deep) neural network (NN) operators is introduced. Density results have been established in the space of continuous functions on [−1,1], with respect to the uniform norm. First, the case of the operators with two-layers is considered in detail, then the definition and the corresponding density results have been extended to the general case of multi-layers operators. All the above definitions allow us to prove approximation results by a constructive approach, in the sense that, for any given f all the weights, the thresholds, and the coefficients of the deep NN operators can be explicitly determined. Finally, examples of activation functions have been provided, together with graphical examples. The main motivation of this work resides in the aim to provide the corresponding multi-layers version of the well-known (shallow) NN operators, according to what is done in the applications with the construction of deep neural models.