a strong influence on the approximation error. In particular, if $\Delta$ is bounded from below independently from $n$, then the approximation error is at most of the order of ${\varepsilon}^2$, while it can only be controlled by ${\varepsilon}$ if $\Delta \lesssim {\varepsilon}$. This key role of $\Delta$ is a consequence of the margin behavior analyzed in Proposition \[thm:margin\] (Section \[s:margin\]) and will also appear in the estimation error.

[**A smoothness assumption.**  ]{} When $d \in {\mathbb{N}}^*$ is fixed, we can minimize the bound of Lemma \[prop:CO\] in ${\varepsilon}$. Unsurprisingly the resulting bound involves the distances $ \| f-f_d\|$ and $\| g-g_d \| $ of $f$ and $g$ to their projections $f_d$ and $g_d$. In the sequel, we assume that the functions $f$ and $g$ are smooth in that their (Fourier) coefficients w.r.t. the basis $(\varphi_j)_{j {\geqslant}1}$ decay sufficiently fast. More precisely, we assume that, for some parameters $s,R>0$, the functions $f$ and $g$ belong to the set $$\label{def:Hs}
{\mathcal{H}_s}(R) := \left\{h \in {\mathbb{L}^2}([0,1]) : \; \sum_{j=1}^{+\infty} c_j(h)^2 j^{2s} {\leqslant}R^2 \right\} \,.$$ The set ${\mathcal{H}_s}(R)$ corresponds to a class of smooth functions with smoothness parameter $s$: when $s=0$, we simply obtain the ${\mathbb{L}^2}([0,1])$-ball of radius $R$. For larger $s$, for example 