The main result of this paper is a constructive proof of a formula
for the upper bound of the approximation error in L_{\infty}
(supremum norm) of multi-dimensional functions by feedforward
networks with one hidden layer of sigmoidal units and a linear
output. This result is applied to formulate a new method of neural
network synthesis. The result can also be used to estimate
complexity of the maximum-error network and/or to initialize that
network weights. An example of the network synthesis is given.

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**Last modified January 20, 2014.**