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

We consider causal time-invariant nonlinear input-output maps that take a s et of locally pth-power integrable functions into a set of real-valued func tions, and we give criteria under which these maps can be uniformly approxi mated arbitrarily well using a certain structure consisting of a not-necess arily-linear dynamic part, followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc, In our results, certai n separation conditions, of the kind associated with the Stone-Weierstrass theorem, play a prominent role. Here they emerge as criteria for approximat ion, 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 th e type addressed can be uniformly approximated arbitrarily well by certain doubly finite Volterra-series approximants if and only if these maps have a pproximately finite memory and satisfy certain continuity conditions.