lower bounds will be provided in Section \[s:lower\].

A classifier in a finite-dimensional setting {#sec:pluginFinitedim}
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Our classifier—defined in Section \[sec:defclassifier\] below—involves a projection step with coefficients $\theta_j$ and $\mu_j$ introduced in Section \[sec:HilbertWhitenoise\] and estimated in Section \[sec:estimatethetamu\].

### ${\mathbb{L}^2}$ orthonormal basis $(\varphi_j)_{j {\geqslant}1}$ and white noise model {#sec:HilbertWhitenoise}

*Orthonormal basis.* Let $(\varphi_j)_{j \in \mathbb{N}^*}$ be any orthonormal basis of ${\mathbb{L}^2}([0,1])$, and $d \in \mathbb{N}^*$ be some dimension that will be chosen as a function of the size $n$ of the sample (for projection purposes). In the sequel, the coefficients $(c_j(h))_{j {\geqslant}1}$ of any function $h \in {\mathbb{L}^2}([0,1])$ w.r.t. the basis $(\varphi_j)_{j {\geqslant}1}$ are defined by $$c_j(h) := \langle \varphi_j, h \rangle = \int_{0}^{1} h(s) \varphi_j(s) ds \,, \qquad j {\geqslant}1 \,,$$ and its ${\mathbb{L}^2}$-projection onto $\mathrm{Span}\left(  \varphi_j , \ 1 {\leqslant}j {\leqslant}d  \right)$ is given by $$\label{eq:defPid}
\Pi_d(h) = \sum_{j=1}^d c_j(h) \varphi_j \,.$$ In particular, we will pay a specific attention to the coefficients of $f$ and $g$ involved in , $$\label{eq:defckfckg}
\forall j {\geqslant}1, \qquad 
\theta_j := c_j(f) \qquad \text{and} \qquad \mu_j:= c_j(g) \,,$$ and to their $d$-dimensional projections $f_d := \Pi_d(f) = \sum_{j=1}^d \theta_j \varphi_j$ and $g_d:=\Pi_d(g) = \sum_{j=1}^d \mu_j \varphi_j$.

 \
*White noise model.* We 