SEPARATION CONDITIONS AND APPROXIMATION OF DISCRETE-TIME AND DISCRETE-SPACE SYSTEMS

We first consider causal time-invariant nonlinear input-output maps th at take a set of bounded functions into a set of real-valued functions , and we give criteria under which these maps can be uniformly approxi mated arbitrarily well using a certain structure consisting of a not n ecessarily linear dynamic part followed by a nonlinear memoryless sect ion that may contain sigmoids or radial basis functions, etc. In our r esults certain separation conditions of the kind associated with the S tone-Weierstrass theorem play a prominent role. Here they emerge as cr iteria for approximation, and not just as sufficient conditions under which an approximation exists. As an application of the results, we sh ow that system maps of the type addressed can be uniformly approximate d arbitrarily well by doubly finite Voltena-series approximants if and only if these maps have approximately finite memory and satisfy certa in continuity conditions. Corresponding results are then given for (no t necessary causal) multivariable input-output maps. Such multivariabl e maps are of interest in connection with image processing.