Criteria for the approximation of nonlinear systems

Conditions for the approximation of discrete-time time-invariant nonlinear systems that act between bounded real sequences are considered. It has been shown in different contexts that if the output of a nonlinear system at each moment is dependent on the remote past of the input only to an arbitrary small extent in a certain sense, then the system can be uniformly approximated arbitrarily well by the maps of certain simple structures such as lattice-map structures, finite Volterra-series structures, dynamic multilayered neural networks with sigmoidal hidden units, and dynamic radial-basis-function networks. Results are given to the effect that, for a certain broad class of system approximation problems, conditions introduced earlier and certain variations of other conditions are all equivalent, and are in fact necessary as well as sufficient