Nonlinear systems and input-output representations

Summary form only given, as follows. Nonlinear and multidimensional extensions of earlier recent theorems by the author are considered. Those results concern the representation of linear operators defined on a set of locally integrable inputs. They establish (for the first time) the existence of impulse responses and associated integral representations as well as the convergence of a classical sum to an integral, in a general setting in which a certain continuity condition plays a central role. They also provide necessary and sufficient conditions for a causal linear operator to have a convolution representation with a continuous impulse-response function. In particular, a representation theorem for nonlinear input-output maps is given that yields, as a special case, Volterra-series representations.<>