Multidimensional nonlinear systems and structure theorems

Nonlinear maps that are shift invariant and have approximately finite memory in a certain very reasonable sense are considered. These maps, which are assumed to satisfy a mild continuity condition, take a subset S of R into R, where R is the set of real-valued maps defined on Zⁿ (as usual, Z is the set of integers and n is an arbitrary positive integer). Results show that all such maps can be approximated arbitrarily well, in the sense of uniform approximation, by the maps of certain special nonlinear structures. This is of interest in connection with nonlinear filtering, system identification, and the general theory of nonlinear systems