An optimal linear filter for random signals with realisations in a separable Hilbert space

Abstract

Let u be a random signal with realisations in an infinitedimensional vector space X and v an associated observable random signal with realisations in a finitedimensional subspace Y X. We seek a pointwisebest estimate of u using a bounded linear filter on the observed data vector v. When x is a finitedimensional Euclidean space and the covariance matrix for v is nonsingular, it is known that the best estimate Ou of u is given by a standard matrix expression prescribing a linear meansquare filter. For the infinitedimensional Hilbert space problem we show that the matrix expression must be replaced by an analogous but more general expression using bounded linear operators. The extension procedure depends directly on the theory of the Bochner integral and on the construction of appropriate Hilbert Schmidt operators. An extended example is given.