$s=1$, we obtain a smaller Sobolev space of functions such that $f' \in {\mathbb{L}^2}([0,1])$ with $\|f'\|_2 {\leqslant}R$.

 \
Under the above assumption on the tail of the spectrum of $f$ and $g$, the loss of accuracy induced by the projection step is easy to quantify. Indeed, for all $f \in {\mathcal{H}_s}(R)$ we have $$\|f-f_{d}\|^2 = \sum_{j=d+1}^{+\infty} c_j(f)^2 {\leqslant}d^{-2s} \sum_{j=d+1}^{+\infty} c_j(f)^2 j^{2s} {\leqslant}R^2 d^{-2s} \,,$$ so that, omitting logarithmic factors, ${\varepsilon}$ can be chosen of the order of $R d^{-s}$ in the statement of Lemma \[prop:CO\].

Estimation error {#sec:estimationerror}
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We now upper bound the estimation error $\mathcal{R}_{f,g}({\widehat}{\Phi}_{d}) - \mathcal{R}_{f,g}(\Phi_d^{\star})$ of our classifier ${\widehat}\Phi_d$. To that end, we first reinterpret $\eta_d$ and $\Phi_d^{\star}$; this will be useful to rewrite the estimation error as an excess risk (as in ) in the truncated space. The next remark follows from direct calculations.

\[rmk:truncatedSpace\] Denote by ${\mathbf{X}}_d := \bigl(\langle\varphi_j,X\rangle\bigr)_{1 {\leqslant}j {\leqslant}d}$, ${\boldsymbol{\theta}}_d = (\theta_j)_{1 {\leqslant}j {\leqslant}d}$, and ${\boldsymbol{\mu}}_d = (\mu_j)_{1 {\leqslant}j {\leqslant}d}$ the versions of $X$, $\theta$, and $\mu$ in the truncated space. Then, $$\eta_d(X) = \frac{\frac{1}{2} q_{f_d}(X)}{\frac{1}{2} q_{f_d}(X) + \frac{1}{2} q_{g_d}(X)} \qquad \textrm{and} \qquad q_{f_d}(X) = e^{\frac{1}{2} \|{\mathbf{X}}_d\|^2} e^{-\frac{1}{2}\|{\mathbf{X}}_d-{\boldsymbol{\theta}}_d\|^2} \;.$$ Since the conditional distribution of ${\mathbf{X}}_d$ is ${\mathcal{N}}({\boldsymbol{\theta}}_d,I_d)$ given $Y=1$ and ${\mathcal{N}}({\boldsymbol{\mu}}_d,I_d)$ given 