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 con-structive 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 exam-ples. 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.