Approximation capability in C(R~/sup n/) by multilayer feedforward networks and related problems

In this paper, we investigate the capability of approximating functions in C(R (n)) by three-layered neural networks with sigmoidal function in the hidden layer. It is found that the boundedness condition on the sigmoidal function plays an essential role in the approximation, as contrast to continuity or monotonity condition. We point out that in order to prove the neural network in the n-dimensional case, all one needs to do is to prove the case for one dimension. The approximation in L(p)-norm (1<p<infinity) and some related problems are also discussed.