Separation conditions and approximation of continuous-time approximately finite memory systems

We consider causal time-invariant nonlinear input-output maps that take a set of locally pth-power integrable 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, not just sufficient conditions under which an approximation exists. As an application of the results and for p=2 we show that system maps of the type addressed can be uniformly approximated arbitrarily well by certain doubly finite Volterra-series approximants if and only if these maps have approximately finite memory and satisfy certain continuity conditions