Nonlinear input-output maps and approximate representations

An approximation theorem is given for causal time-invariant nonlinear maps that take one set of functions defined on [0, ∞) into another. The theorem is used to show that, under some typically very reasonable conditions, an input-output map can be approximated arbitrarily well in a meaningful sense by a finite Volterra series, even though it may not have a Volterra series expansion. The set of inputs on which the approximation holds need not be compact, and the inputs need not be continuous.