distance. In the literature, $\mathbb{L}_2$ tomography is usually performed for boundedly supported functions, see  and . However, most Wigner function do not have a bounded support! Instead, we use the fact that Wigner functions in the class $\mathcal{R}(B,r)$ decrease very fast and show that a properly truncated estimator attains the rates we may expect from the statistical problem of deconvolution in presence of tomography. Thus, we modify the estimator by truncating it over a disc with increasing radius, as $n \to \infty$. Let us denote $$D(s_n) = \left\{z=(q,p) \in \mathbb{R}_2: \|z\| \leq s_n \right\},$$ where $s_n \to \infty$ as $n \to \infty$ will be defined in Theorem \[minimax.l2.bounds\]. Let now $$\label{truncated.estimator}
    \widehat W^{\eta, *}_{h,n}(z) = \widehat{W} ^\eta_{h,n} (z) I_{D(s_n)}(z).$$ From now on, we will denote for any function $f$, $$\|f\|^2_{D(s_n)}= \int_{D(s_n)} f^2 (z) dz ,$$ and by $\overline D(s_n)$ the complementary set of $D(s_n)$ in ${\mathbb R}^2$. Then, $$\begin{aligned}
    E\left[\left\|\widehat W_{h}^{\eta,*} - W_{\rho} \right\|_2^2\right]
    &=& E\left[\left\|\widehat W_{h}^\eta - W_{\rho} \right\|_{D(s_n)}^2
    \right] + \|W_\rho\|^2_{\overline D(s_n)} \\
    &=& E\left[\left\|\widehat W_{h}^\eta -
            E\left[\widehat{W}_{h}^\eta\right] \right\|_{D(s_n)}^2 \right]
    +\left\|E\left[\widehat W_{h}^\eta\right] - W_{\rho}
    \right\|_{D(s_n)}^2 \\
    && +\|W_\rho\|^2_{\overline D(s_n)}.\end{aligned}$$ When replacing the $\mathbb{L}_2$ norm with the above restricted integral, the upper bound of the bias of the estimator 