Multivariate Neural Networks over infinite domain as Positive Linear Operators and a touch of measure theory

Multivariate neural network operators of infinite domain are interpreted as positive linear operators and related general theory applies to them. These operators are induced by a symmetrized density function deriving from the parametrized and deformed hyperbolic tangent activation function. We are acting on the space of continuous and bounded functions on RN, N∈ N, to the reals. We study quantitatively the rate of convergence of these multivariate neural network operators to the unit operator. Our inequalities involve the modulus of continuity of the function under approximation or its Fr\'{e}chet derivative. Connection to measure theory is shown.