Neural Networks as Positive Linear Operators

Basic neural network operators are interpreted as positive linear operators and the 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. I explore the space of continuous functions on a compact interval of the real line to the reals. I study quantitatively the rate of convergence of these neural network operators to the unit operator. The studied inequalities involve the modulus of continuity of the function under approximation or its derivative. I produce uniform and Lp, p≥1, approximation results via these inequalities. The convexity of functions is also taken into consideration.