AN OPTIMAL LINEAR FILTER FOR ESTIMATION OF RANDOM FUNCTIONS IN HILBERT SPACE
Keyword(s):
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.
Keyword(s):
1965 ◽
Vol 17
◽
pp. 1030-1040
◽
Keyword(s):
1984 ◽
Vol 27
(2)
◽
pp. 229-233
◽
2018 ◽
Vol 64
(1)
◽
pp. 194-210
1957 ◽
Vol 53
(2)
◽
pp. 304-311
◽