R₊ fading memory and extensions of input-output maps

Much is known about time-invariant nonlinear systems with inputs and outputs defined on R⁺ that possess approximately finite memory. For example, under mild additional conditions, they can be approximated arbitrarily well by the maps of certain interesting simple structures. An important fact that gives meaning to results concerning such systems is that the approximately finite memory condition is known to be often met. Here we consider the known proposition that if a causal time-invariant continuous-time input-output map H has fading memory on a certain set of bounded functions defined on all of R, then H can be approximated arbitrarily well by a finite Volterra series operator. We show that in a certain sense, involving the existence of extensions of system maps, this result too has wide applicability.