On estimation and detection of infinite-variable function

We observe an unknown infinite-variable function $f=f(t)$, $t=(t_1,\ldots,t_n,\ldots)\in[0,1]^\infty$, in the white Gaussian noise of a level $\varepsilon>0$. We suppose that, in each variable, there exist 1-periodical $\sigma$-smooth extensions of functions $f(t)$ on $\mathbb R^\infty$. Taking a quantity $\sigma>0$ and a positive sequence $\mathbf a=\{a_k\}$, we consider the set $\mathcal F_{\sigma,\mathbf a}$ that consists of functions $f$ such that $\sum_{k=1}^\infty a_k^2\|\partial^\sigma f/\partial t_k^{\sigma}\|_2^2\le 1$. We consider the cases $a_k=k^\alpha$ and $a_k=\exp(\lambda k)$, $\alpha>0$, $\lambda>0$. We want to estimate a function $f\in\mathcal F_{\sigma,\mathbf a}$ or to test the null hypothesis $H_0$: $f=0$ against alternatives $f\in\mathcal F_{\sigma,\mathbf a}(r_\varepsilon)$ where the set $\mathcal F_{\sigma,\mathbf a}(r)$ consists of functions of $f\in \mathcal F_{\sigma,\mathbf a}$ such that $\|f\|_2\ge r$.
In the estimation problem, we obtain the asymptotics (as $\varepsilon\to 0$) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates $r_\varepsilon^*$ that provide distiguishability in the problems.