Convergence of a family of neural network operators of the Kantorovich type

A family of neural network operators of the Kantorovich type is introduced and their convergence studied. Such operators are multivariate, and based on certain special density functions, constructed through sigmoidal functions. Pointwise as well as uniform approximation theorems are established when such operators are applied to continuous functions. Moreover, also $L^p$ approximations are considered, with $1 \miu p < +\infty$, since the $L^p$ setting is the most natural for the neural network operators of the Kantorovich type. Constructive multivariate approximation algorithms, based on neural networks, are important since typical applications to neurocomputing processes do exist for high-dimensional data, then the relation with usual neural networks approximations is discussed. Several examples of sigmoidal functions, for which the present theory can be applied are presented.