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...

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Bibliographic Details
Published in:The ANZIAM journal 2020-07, Vol.62 (3), p.274-301
Main Authors: HOWLETT, PHIL, TOROKHTI, ANATOLI
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Eigenvalues
Hilbert space
Linear filters
Linear operators
Mean square errors
Operators (mathematics)
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Summary: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.
ISSN:1446-1811
1446-8735
DOI:10.1017/S1446181120000188