Bounds for the risks of nonparametric estimates of the regression

Let us assume that the observations $Y_1,\dots,Y_N$ have the form (0.1) and that it is known only that $f$ belongs to the set $\Sigma$ of $2\pi$-periodical functions in some functional space. We consider the loss function of the type $l(\|\hat f_N-f\|_\infty)$, where $l(x)$ increases for $x>0$, and prove that the equidistant experimental design and the estimator (1.4) for $f$ are asymptotically optimal in the sense of the rate of convergence of risks for the wide class of sets $\Sigma$ if the integer $n$ in (1.4) satisfies the equation (1.14). In particular, the optimal order of the rate of convergence is $(N/\ln N)^{-\beta/(2\beta+1)}$ if $\Sigma$ is the set of periodical functions with smoothness $\beta$.