The uniqueness theorem for analytic functions. Statistical approach

An analytic function $f$ is fully determined by its values at any arbitrary small domain $D$. The problem of determination of the analytic function $f$ outside $D$ is an ill-posed problem, and small errors in the determination $f$ in $D$ may give rise to great errors of the determination $f$ outside $D$.
In this talk we consider the following problem. An entire function $f$ is observed in a small noise on a set $D$. How far from $D$ one can determine $f$ with small errors? A simple variant of the problem is the following one. A function $f$ is observed on an interval $[a;b]$ in the additive Gaussian noise of intensity $\varepsilon$. For example, the observation
$$ X(t)=\int_0^tf(u)\,du+\varepsilon w(t), \qquad w(a)=0, $$
$w(t)$ is a Wiener process. It is supposed that $f$ belongs to an a priori known set of entire functions whose order of growth is at most $\rho$. It turns out that the the consistent estimation of $f$ is possible on the distances of the order $(\ln(\varepsilon^{-1}))^{1/\rho}$ and is impossible on the distances of larger order. The same results can be proved for other variants of similar problems.