Approximately finite memory and the circle criterion

In recent work a complete characterization has been given of those input-output maps G that can be uniformly approximated by the maps of certain simple structures. The criterion is that G must satisfy certain continuity and approximately finite memory conditions. It is proved here that the conditions are satisfied by the input-output maps of feedback systems of a familiar type containing a (possibly distributed) linear part and 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, can be input-output approximated arbitrarily well by a structure that takes the form of a feedforward dynamical neural network