Replacing points by compacta in neural network approximation

It is shown that cartesian product and pointwise-sum with a fixed compact set preserve various approximation-theoretic properties. Results for pointwise-sum are proved for F-spaces and so hold for any normed linear space, while the other results hold in general metric spaces. Applications are given to approximation of Lp-functions on the d-dimensional cube, 1⩽p<∞, by linear combinations of half-space characteristic functions; i.e., by Heaviside perceptron networks.