Uniform approximation and the circle criterion

The input-output maps G of causal time-invariant systems are considered, and a complete characterization of those maps that can be uniformly approximated by the maps of certain simple structures is given. The criterion is that G must satisfy certain continuity and approximately-finite-memory conditions. It is proved that the conditions are satisfied by the input-output maps of control systems of a familiar type containing a sector nonlinearity for which the circle condition for stability is met. In particular, this shows that such feedback systems, with inputs drawn from a certain large set of bounded functions, possess arbitrarily good finite Volterra-series (or radial-basis-function) approximations