Wavelet-based Fock space: a new multi-scale space for nonlinear dynamical systems

Generalized Fock (GF) spaces were introduced by de Figueiredo and associates in late 1970's and early 1980's for generic representation of the input-output maps of nonlinear dynamical systems. Since then the underlying concepts and methods have been used and further developed by the present authors and others in the context of a number of applications including neural networks (see, e.g., the paper by de Figueiredo in the invited session on Fundamental of Neural Networks in this ISCAS'96 Proceedings). A GF Space F consists of sequences of tensor products of a given Hilbert space H. When H is L₂(R), the elements of F are Volterra series. In the present paper we introduce a GF space F whose elements are constructed from an L₂(R) equipped with an orthonormal wavelet basis. This provides a unique setting for modeling identification and control of nonlinear dynamical systems at multiple scales as elicited by the underlying wavelet basis.