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

Authors

DOI:

https://doi.org/10.21914/anziamj.v62.15576

Keywords:

random functions, optimal estimation, linear operators, generalized inverse operators

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

Author Biographies

Phil George Howlett, University of South Australia

Emeritus Professor, Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, South Australia, Australia.

Anatoli Torokhti, University of South Australia

Associate Professor, Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, South Australia, Australia.

Published

2021-02-04

Issue

Section

Articles for Printed Issues