In the first of this dissertation we consider the problem of rational approximation and identification of stable linear systems. Affine wavelet decompositions of the Hardy space H2 (II+), are developed as a means of constructing rational approximations to nonrational transfer functions. The decompositions considered here are based on frames constructed from dilations and complex translations of a single rational function. It is shown that suitable truncations of such decompositions can lead to low order rational approximants for certain classes of time-frequency localized systems. It is also shown that suitably truncated rational wavelet series may be used as 'linear-in-parameters' black box models for system identification. In the context of parametric models for system identification, time-frequency localization afforded by affine wavelets is used to incorporate a priori knowledge into the formal properties of the model. Comparisons are made with methods based on the classical Laguerre filters.

The second part of this dissertation is concerned with developing a theoretical framework for feedforward neural networks which is suitable for both analysis and synthesis of such networks. Our approach to this problem is via affine wavelets and the theory of frames. Affine frames for L2, are constructed using combinations of sigmoidal functions and the inherent translations and dilations of feedforward network architectures. Time- frequency localization is used in developing methods for the synthesis of feedforward networks to solve a given problem.

These two seemingly disparate problems both lie within the realm of approximation theory, and our approach to both is via the theory of frames and affine wavelets.