\sqrt2(ip-q) \right)}}^{m-n}
    L_n^{m-n}{{\left( 2q^2+2p^2 \right)}}
    \label{eq:WmnLaguerre}\end{aligned}$$ thus, writing $z {\mathrel{\mathop:} = }\sqrt{q^2+p^2} $, $$\label{eq:deflmn}
    l_{m,n}(z) {\mathrel{\mathop:} = }{{\left\lvertW_{m,n}(q,p)\right\rvert}} = \frac{2^{\frac{m-n}2}}{\pi}
    {{\left( \frac{n!}{m!} \right)}}^\frac12 e^{-z^2} z^{m-n}
    {{\left\lvertL_n^{m-n}(2z^2)\right\rvert}}$$ where $L_{n}^{\alpha}(x) {\mathrel{\mathop:} = }(n!)^{-1} e^x x^{-\alpha} \frac{d^n}{dx^n}
(e^{-x} x^{n+\alpha})$ is the Laguerre polynomial of degree $n$ and order $\alpha$. Concerning the Fourier transforms, we also recall that $${\widetilde{W_{m,n}}}(q,p) = \frac{(-i)^{m+n}}{2}
    {W_{m,n}}{{\left( \frac{q}{2},\frac{p}{2} \right)}}.
    \label{eq:FourierWmn}$$

In this section we show how a decrease condition on the coefficients of the density matrix translates on the corresponding Wigner distribution. First the case $r < 2$:

\[prop:popetown\] Assume that $0 < r < 2$ and that there exists $B > 0$ such that, for all $m \geq n$, $${{\left\lvert\rho_{m,n}\right\rvert}} \leq e^{-B(m+n)^{r/2}}.  \nonumber$$ Then for all $\beta < B$, there exists $z_{0}$ (depending explicitly on $r, B, \beta$, see proof) such that $z := \sqrt{q^2 + p^2} \geq z_{0}$ implies $$\label{eq:dec}
        {{\left\lvertW_{\rho}(q,p)\right\rvert}} \leq A(z) e^{- \beta z^r}$$ as well as $$\label{eq:reg}
        {{\left\lvert{\widetilde{W_{\rho}}}(q,p)\right\rvert}} \leq A(z/2) e^{- \beta (z/2)^r}$$ where $A(z) {\mathrel{\mathop:} = }\frac1\pi {{\left( \sum\limits_{m,n} e^{-B(m+n)^{r/2}} +
      \frac{4}{B r} z^{4-r} \right)}}$.

If $r=2$, the result is a little different:

\[prop:popetown2\] Suppose that there exists $B > 0$ such that, for all $m \geq n$, $${{\left\lvert\rho_{m,n}\right\rvert}} \leq e^{-B(m+n)}.  \nonumber$$ Then 