Applications of reproducing kernels, and state space methods to signal processing, and wavelets

The aim of this multidisciplinary project was to pose new problems and develop new methods at the interface between mathematics and signal processing, by bringing together the competence of the four Pl’s from different disciplines: Classical operators and complex analysis, Mathematical System Theory, Signal Processing and Electrical Engineering. The effort turned to be rewarding. During the four years of the grant, we explored the interplay between functional analysis, stochastic processes, and filters, via the notion of positive definite functions. We used state space methods, functional analysis (and in particular topological vector spaces, reproducing kernel Hilbert spaces) The results stemming out of this research include a description of all wavelet filters using state space method, and the study of underlying multiscale systems. This in turn has lead us to posing and solving new kind of interpolation problems in the setting of analytic functions. On another direction we applied white noise analysis methods to study stochastic distributions and new classes of important generalzied stochastic processes. Finally we studied discrete analytic functions and new aspects of the theory of reproducing kernel spaces.