Backpropagation representation theorem using power series

A representation theorem is developed for backpropagation neural networks. First, it is assumed that the function to be approximated,
F
(
x
) for the vector
x
, is continuous and has finite support, so that it can be approximated arbitrarily well by a multidimensional power series. The activation function, sigmoid or otherwise, is then approximated by a power-series function of the net. Basic building-block subnetworks, realizing the monomial or product of the inputs, are implemented with any desired degree of accuracy. Each term in the power series for
F
(
x
) is realizable using a building block, and each building block has one hidden layer. Hence, the overall network has one hidden layer