Approximation results for neural network operators activated by sigmoidal functions.

In this paper, we study pointwise and uniform convergence, as well as the order of approximation, for a family of linear positive neural network operators activated by certain sigmoidal functions. Only the case of functions of one variable is considered, but it can be expected that our results can be generalized to handle multivariate functions as well. Our approach allows us to extend previously existing results. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of neural network operators activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. In particular, we show that for C(1)-functions, the order of approximation for our operators with logistic and hyperbolic tangent functions here obtained is higher with respect to that established in some previous papers. The case of quasi-interpolation operators constructed with sigmoidal functions is also considered.