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

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/S144618112000018