Asymptotic expansions for the neural network operators of the Kantorovich type and high order of approximation

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the density functions generated by suitable sigmoidal functions. Furthermore, to improve the rate of convergence, we consider finite linear combinations of the above neural network type operators, and also in the latter case, we obtain a Voronovskaja type theorem. Finally, concrete examples of sigmoidal activation functions have been deeply discussed, together with the case of rectified linear unit (ReLu) activation function, very used in connection with deep neural networks.