Network approximation of input-output maps and functionals

We give results concerning the problem of approximating the input-output maps of nonlinear discrete-time approximately-finite-memory systems. Here the focus is on the linear dynamical parts of the approximating structures, and we give examples showing that these linear parts can be derived from a single prespecified function that meets certain conditions. This is done in the context of an approximation theorem in which attention is focused on what we call "basic sets". We also consider the related but very different problem of approximating nonlinear functionals using lattice operations or the usual linear ring operations. For this problem we give criteria, not just sufficient conditions, for approximation on compact subsets of reflexive Banach spaces.