In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure theoretic condition for density of such networks in the space of continuous functions. Further, we prove a density result for neural networks with a specifically constructed activation function and a fixed number of neurons.

翻译：本文研究权重在有限个方向变化且阈值取自开区间的单隐层神经网络的逼近性质。我们得到了此类网络在连续函数空间中稠密性的一个必要且同时充分的测度论条件。进一步，我们证明了具有特定构造激活函数和固定神经元数量的神经网络的稠密性结果。