希尔伯特空间中随机函数估计的最优线性滤波器
An optimal linear filter for estimation of random functions in Hilbert space
摘要：
令${\mbox{$ \mbox{\boldmath $f$} $}}$是在希尔伯特空间$H$中具有可观测实现的平方可积、零均值随机向量，并且令${\mbox{$ \mbox{\boldmath $g$} $}}$是在希尔伯特空间$K$中具有不可观测实现的相关平方可积、零均值随机向量。我们寻求一个最佳的过滤器的形式的封闭的线性算子$X$作用于一个接近的矢量${\mbox{$ \mbox{\boldmath $f$} $}}_ε \approx {\mbox{$ \mbox{\boldmath $f$} $}}$的可观察的实现，提供最佳的估计$\widehat{{\mbox{$ \mbox{\boldmath $g$} $}}}_ε = X {\mbox{$ \mbox{\boldmath $f$} $}}_ε$的矢量${\mbox{$ \mbox{\boldmath $f$} $}}$。我们假设所需的协方差算子是已知的。最后给出了一个典型算例。
英文摘要：
Let ${\mbox{$\mbox{\boldmath $f$}$}}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space $H$, and let ${\mbox{$\mbox{\boldmath $g$}$}}$ 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 ${\mbox{$\mbox{\boldmath $f$}$}}_ε \approx {\mbox{$\mbox{\boldmath $f$}$}}$ that provides the best estimate $\widehat{{\mbox{$\mbox{\boldmath $g$}$}}}_ε = X {\mbox{$\mbox{\boldmath $f$}$}}_ε$ of the vector ${\mbox{$\mbox{\boldmath $f$}$}}$. We assume the required covariance operators are known. The results are illustrated with a typical example.
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