Separation conditions and approximation of discrete-time and discrete-space systems

Abstract

We first consider causal time-invariant nonlinear input-output maps that take a set of bounded functions into a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not necessarily linear dynamic part followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc. In our results certain separation conditions of the kind associated with the Stone-Weierstrass theorem play a prominent role. Here they emerge as criteria for approximation, and not just as sufficient conditions under which an approximation exists. As an application of the results, we show that system maps of the type addressed can be uniformly approximated arbitrarily well by doubly finite Volterra-series approximantsif and only if these maps have approximately finite memory and satisfy certain continuity conditions. Corresponding results are then given for (not necessary causal) multivariable input-output maps. Such multivariable maps are of interest in connection with image processing.