Separation conditions and criteria for uniform approximation of input-output maps

Abstract

We consider multidimensional shift-invariant input-output maps, 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 sufficient conditions under which an approximation exists.