An optimal linear filter for estimation of random functions in Hilbert space
Keyword(s):
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
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1965 ◽
Vol 17
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pp. 1030-1040
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1984 ◽
Vol 27
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pp. 229-233
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2018 ◽
Vol 64
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pp. 194-210
1957 ◽
Vol 53
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pp. 304-311
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