Doubly-finite Volterra-series Approximations

We consider causal time-invariant nonlinear inputoutput maps that take a set of bounded functions into a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a notnecessarily linear dynamic part followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc. As an application of the results, we show that system maps of the type addressed can be uniformly approximated arbitrarily well by doublynite Volterra-series approximants if and only if these maps have approximatelynite memory and satisfy certain continuity conditions. Corresponding results have also been obtained for (not necessary causal) multivariable input-output maps. Such multivariable maps are of interest in connection with image processing.