Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
Observing an unknown $n$-variables function $f(t), t\in [0,1]^n$ in the white Gaussian noise of a level $\e>0$. We suppose that there exist $1$-periodical (in each variable) $\sigma$-smooth extensions of functions $f(t)$ on $\R^n$ and $f$ belongs to a Sobolev ball, i.e., $\ f\ _{\sigma,2}\leq 1$, where $\ \cdot\ _{\sigma,2}$ is a Sobolev norm (we consider two variants of one). We study two problem: estimation of $f$ and testing of the null hypothesis $H_0: f=0$ against alternatives $\ f\ _2\geq r_\e$. We study the asymptotics (as $\e\to 0,\ n\to\infty$) of the minimax risk for square losses, for estimation problem, and of minimax error probabilities and of minimax separation rates in the detection problem. We show that of $n\to\infty$, then there exist ``sharp separation rates'' in the detection problem. The asymptotics of minimax risks of estimation and of separation rates of testing are of different type for $n\ll \log\e^{-1}$ and for $n\gg \log\e^{-1}$. The problems under consideration are closely related with ``lattice problem'' in the numerical theory.