there exists $z_{0}$ such that $z := \sqrt{q^2 + p^2} \geq
    z_{0}$ implies $$\label{eq:dec2}
        {{\left\lvertW_{\rho}(q,p)\right\rvert}} \leq A(z) e^{- \frac{B}{(1+\sqrt{B})^2}
          z^2}$$ as well as $$\label{eq:reg2}
        {{\left\lvert{\widetilde{W_{\rho}}}(q,p)\right\rvert}} \leq A(z/2) e^{-
          \frac{B}{(1+\sqrt{B})^2} (z/2)^2}$$ for $A(z) = \frac1\pi {{\left( \sum\limits_{m,n} e^{-B(m+n)} + \frac{2
        e^{B}}{B(1+\sqrt{B})^2} z^{2} \right)}}$.

Note that $\frac{B}{{{\left( 1+\sqrt{B} \right)}}^2} < \min(B,1)$. Even when $B$ is very large, we cannot hope to obtain a faster decrease because $e^{-z^2}$ is the decrease rate of the basis functions themselves (Lemma \[lemm:estimeLaguerre\]).

The proof of these propositions is defered to Appendix \[sec.proofs\]. More general results and converses are studied in [@Aubry:2007fk]. Let us now state a few general utility lemmata.

\[lemm:ineqdiff\] Let $y$ and $w$ be two $C^2$ functions: $[x_0,+\infty) \to
    (0,+\infty)$ such that $y'(x) \to 0$, $w$ is bounded, satisfying the differential equations $$\begin{aligned}
        y''(x) &=& \phi(x) y(x) \\
        w''(x) &=&\psi(x) w(x),
    \end{aligned}$$ with continuous $\phi(x) \leq \psi(x)$, and initial conditions $y(x_0) = w(x_0)$. Then for all $x \geq x_0$, $w(x) \leq y(x)$.

Suppose that there exists $x_1 \geq x_0$ where $w(x_1) > y(x_1)$. Then for some $x_2 \in[x_0,x_1]$ we have $w'(x_2) > y'(x_2)$ and $w(x_2) \geq y(x_2)$. Consequently, for all $x \geq x_2$, $w''(x)
    - y''(x) \geq 0$, and $w'(x)-y'(x) \geq w'(x_2) - y'(x_2)$. 