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

Abstract

Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space $H$, and let $\boldsymbol{g}$ 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 $\boldsymbol{f}_{\epsilon} \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X\! \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$. We assume the required covariance operators are known. The results are illustrated with a typical example.

doi:10.1017/S1446181120000188