SEPARATION CONDITIONS AND CRITERIA FOR UNIFORM APPROXIMATION OF INPUT-OUTPUT MAPS

We consider multidimensional shift-invariant input-output maps G from a relatively compact set of functions S to a set of real-valued functi ons, and we give criteria under which these maps can be uniformly appr oximated arbitrarily well using a certain structure consisting of a no t-necessarily linear dynamic part followed by a non-linear memoryless section that may contain sigmoids or radial basis functions, etc. The dynamic part is comprised of a finite number of dynamic maps h(l),..., h(j) drawn from a set H of maps that satisfy a certain continuity cond ition. In our results certain separation conditions, of the kind assoc iated with the Stone-Weierstrass theorem, play a prominent role. Here they emerge as criteria for approximation, and not just sufficient con ditions under which an approximation exists. In particular, one of the theorems given is a result to the effect that universal approximation can be achieved using the structure we consider if and only if the se t H satisfies the separation condition that (hu(1))(0)not equal(hu(2)) (0) for some h is an element of H whenever u(1),u(2) is an element of cl(S) and u(1) not equal u(2) (where cl(S) denotes the closure of S). This holds even if the elements of H are not linear. (C) 1998 John Wil ey & Sons, Ltd.