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IMA Journal of Mathematical Control and Information 1986 3(2-3):223-236; doi:10.1093/imamci/3.2-3.223
© 1986 by Institute of Mathematics and its Applications
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Systems in Reproducing Kernel Hilbert Space: Causality, Realizability, and Separability

CARLA A. SCHWARTZ and BRADLEY W. DICKINSON

Department of Electrical Engineering, McGill University Montreal PQ Canada H3A 2A7
Department of Electrical Engineering, Princeton University Princeton NJ 08544 USA

We describe a method of obtaining finite dimensional linear state space realizations for solutions to a generalized innovations representation problem: to characterize finite dimensional realizability of a causally invertible transformation which relates two given random processes. The solution is determined on a reproducing kernel Hilbert Space, (RKHS), where the reproducing kernel is the covariance function of one of the given random processes, u(·). This investigation succeeds in characterizing finite dimensional linear RKHS (innovations) realizations for the other random process, y(·), in terms of u(·), with a separability property of both the reproducing kernel and the covariance of the given random process, y(·).


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