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

TL;DR

This paper develops an optimal linear filtering method in Hilbert spaces for estimating unobservable random vectors from observable data, assuming known covariance operators, and demonstrates its application with an example.

Contribution

It introduces a closed-form optimal linear filter in Hilbert spaces for estimating unobservable random functions based on observable data.

Findings

Derives an explicit form of the optimal filter operator.

Shows the filter's effectiveness through a practical example.

Abstract

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