Neural network operators: Constructive interpolation of multivariate functions

In this paper, the interpolation of multivariate data by operators of the neural network type is proved. These operators can also be used to approximate continuous functions defined on a box-domain of RdRd. In order to show this fact, a uniform approximation theorem with order is proved. The rate of approximation is expressed in terms of the modulus of continuity of the functions being approximated. The above interpolation neural network operators are activated by suitable linear combinations of sigmoidal functions constructed by a procedure involving the well-known central B-spline. The implications of the present theory with the classical theories of neural networks and sampling operators are analyzed. Finally, several examples with graphical representations are provided.