Notes on uniform approximation of time-varying systems on finite time intervals

It is shown that the elements G of a large class of causal input-output maps can be uniformly approximated arbitrarily well using a certain structure if and only if G is uniformly continuous. For the case considered the system inputs and outputs are defined on a finite interval [O,t_f]. Our approximating structure involves certain functions that can be chosen in different ways. For the special case in which these functions are taken to be certain polynomial functions, the input-output map of our structure is a generalized Volterra series