The role of the RKH space F in the analysis and design of recurrent neural networks

The space F(H), or simply F, is a reproducing kernel Hilbert space (RKHS) of analytic (nonlinear) functional (Volterra functional) on a separable Hilbert space H. It was introduced in the late 1970´s by the author, in collaboration with T.A.W. Dwyer, III, and L. Zyla, to represent input-output maps of large scale nonlinear dynamical systems. In the present paper we show how the properties of F, and, in particular, its reproducing kernel, can be used to model the structure and behavior of recurrent neural networks (RNNs).