g- and h-representations for nonlinear maps

AbstractInput-output representation results are given for systems whose inputs and out-puts are real-valued functions on the whole real line R, the half line [0, ∞), or Rm. These results, which deal with noncausal as well as causal maps, considerably extend related previous results which concern causal maps defined on functions on the half line. The results address nonlinear systems whose input-output maps meet certain continuity conditions. They establish the existence of integral representations, which we call “g-representations.” These concern system responses to bounded inputs. Additional conditions are given under which a nonlinear map has an integral representation, called an “h-representation,” for which the integrands have a natural interpretation in terms of impulse responses. These results provide a means of studying the input-output properties of nonlinear systems in terms of the familiar impulse response concept, and they extend the concept of integral transformation to nonlinear maps