Multivariate Neural Network Operators: Simultaneous Approximation and Voronovskaja-Type Theorem

In this paper, the simultaneous approximation and a Voronoskaja-type theorem for the multivariate neural network operators of the Kantorovich type have been proved. In order to establish such results, a suitable multivariate Strang-Fix type condition has been assumed. A crucial step in the established proofs is given by the application of certain auxiliary results (here established) involving the partial derivatives of the considered multivariate density functions. Other than convergence theorems, we also establish quantitative estimates for the order of simultaneous approximation thanks to the use of the modulus of continuity of the target function. Here, sigmoidal, rectified linear unit (ReLu), and rectified power units (RePUs) functions have been considered as activation functions.