Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series

Ahsfract-Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-inuariant (TZ) con~inuou.r nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finiiedimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time inlero& and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that nny TZ operator with fading memory can be approximated (in our strong sense) by a nonlinear movingaverage operator. Some further discussion of the notion of fading memory is given.