Approximation theorems for discrete-time systems

Some strong approximation results concerning discrete-time nonlinear input-output maps are presented. The results concern maps G that are causal, time invariant, and satisfy certain continuity and approximately finite-memory conditions. It is shown that such G´s can be uniformly approximated arbitrarily well by finite sums. This can be used, for example, as a basis for adaptive filtering or for the adaptive identification of G´s. Similar propositions hold also for G´s that are not necessarily causal