An optimal linear filter for estimation of random functions in Hilbert space
Abstract
Let {\mbox{\mbox{\boldmath }}} be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space , and let {\mbox{\mbox{\boldmath }}} be an associated square-integrable, zero-mean, random vector with realizations, which are not observable, in a Hilbert space . We seek an optimal filter in the form of a closed linear operator acting on the observable realizations of a proximate vector {\mbox{\mbox{\boldmath }}}_{\epsilon} \approx {\mbox{\mbox{\boldmath }}} that provides the best estimate \widehat{{\mbox{\mbox{\boldmath }}}}_{\epsilon} = X {\mbox{\mbox{\boldmath }}}_{\epsilon} of the vector {\mbox{\mbox{\boldmath }}}. We assume the required covariance operators are known. The results are illustrated with a typical example.
Cite
@article{arxiv.2008.12485,
title = {An optimal linear filter for estimation of random functions in Hilbert space},
author = {Phil Howlett and Anatoli Totokhti},
journal= {arXiv preprint arXiv:2008.12485},
year = {2020}
}