An optimal linear filter for estimation of random functions in Hilbert space

08/28/2020

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by   Phil Howlett, et al.

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Let be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let be an associated square-integrable, zero-mean, random vector with realizations, which are not observable, in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector _ϵ≈ that provides the best estimate _ϵ = X _ϵ of the vector . We assume the required covariance operators are known. The results are illustrated with a typical example.