Notes on fading-memory conditions

In some studies concerning the approximation of nonlinear systems, the concept of a weighting function w plays a central role. This paper, in which attention is focused on continuous-time cases, directs attention to some interesting properties of systems that have R+ fading memory or fading memory. In particular, we show that half-line input-output maps that have R+ fading memory with respect to some w in fact have R+ fading memory with respect to all such w's. And we show that a similar proposition holds in a setting concerning Volterra-series approximations for continuous-time systems with inputs and outputs defined on all of R. We show also that, in that setting, fading memory is equivalent to uniform fading memory. Some related results are also described.