On estimating functions of the mean

An estimation problem is considered for a function $\varphi(\alpha_1,\dots,\alpha_N)$ of unknown complex parameters $\alpha_1,\dots,\alpha_N$ by observations $\xi(t)=\alpha_1\theta_1(t)+\dots+\alpha_N\theta_N(t)+\Delta(t)$, $t\in T$, where $\Delta(t)$ is complex Gaussian stochastic function.
The main result is: the best unbiased estimate of an analytic function $\varphi(\alpha_1,\dots,\alpha_N)$ is $\varphi(\widehat\alpha_1,\dots,\widehat\alpha_N)$ where $\widehat\alpha_k$ are the BLUE of regression coeffitients $\alpha_k$. The real-valued case and the case of infinite dimensional regression are briefly discussed.