On the approximation by single hidden layer feedforward neural networks with fixed weights

Single hidden layer feedforward neural networks (SLFNs) with fixed weights possess the universal approximation property provided that approximated functions are univariate. But this phenomenon does not lay any restrictions on the number of neurons in the hidden layer. The more this number, the more the probability of the considered network to give precise results. In this note, we constructively prove that SLFNs with the fixed weight 1 and two neurons in the hidden layer can approximate any continuous function on a compact subset of the real line. The proof is implemented by a step by step construction of a universal sigmoidal activation function. This function has nice properties such as computability, smoothness and weak monotonicity. The applicability of the obtained result is demonstrated in various numerical examples. Finally, we show that SLFNs with fixed weights cannot approximate all continuous multivariate functions.