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 p^th-power integrable functions into a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated 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. 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