L^p simultaneous approximation by neural networks with one hidden layer

It is shown in this paper by a constructive method that for any Lebesgue integrable functions defined on a compact set in a multidimensional Euclidian space, the function and its derivatives can be simultaneously approximated by a neural network with one hidden layer. This approach naturally yields the design of the hidden layer and the convergence rate. The obtained results describe the relationship between the rate of convergence of networks and the numbers of units of the hidden layer, and generalize some known density results in uniform measure.