New general nonlinear representations for system operators

A representation for nonlinear system operators is proposed by considering a nonlinear dynamic system as a continuous time-dependent functional for causal signals. The representation is based on a Gaussian-type kernel functional and is suitable for some special purposes such as nonlinear system identification, pattern recognition, nonlinear estimation, and signal processing. It is proved that the basis of such a representation is a generalized Haar system, and under certain very weak conditions on the coefficients of the resultant system, it is shown that the reconstructed nonlinear system converges to a continuous time-dependent functional which represents a true system, as the number of input-output data-points tends to infinity. An existence and uniqueness result on the best uniform approximation of such finite-dimensional representations to a given system is established.