Radon transform and differentiable approximation by neural networks

We treat the problem of simultaneously approximating C/sup m/-functions in several variables and their derivatives by superpositions of a fixed activation function in one variable. The domain of approximation can be either compact subsets or the whole Euclidean space. If the domain is compact, the activation function does not need to be scalable. Even if the domain is the whole space, the activation function can be used without scaling under a certain condition. The approximation can be implemented by a three layered neural network with hidden layer units having the activation function.