be expressed as kernel integrals: for all $j,k
\in {\mathbb{N}}$, $$\begin{aligned}
    \label{rhojk}
    \rho_{j,k}=\frac 1\pi\int \int_0^\pi
    p_\rho(x,\phi)f_{j,k}(x)e^{-i(k-j)\phi}d\phi dx\end{aligned}$$ where $f_{j,k} = f_{k,j}$ are bounded real functions called *pattern functions* in quantum homodyne literature. A concrete expression for their Fourier transform using Laguerre polynomials was found in [@Richter1]: for $j \geq k$, $$\begin{aligned}
    \tilde{f}_{k,j}(t) &=& 2 \pi^2 {{\left\lvertt\right\rvert}} \widetilde{W_{j,k}}(t,0)
    \nonumber
    \\   \label{pattern}
    &=&\pi(-i)^{j-k}\sqrt{\frac{2^{k-j} k!}{j!}}|t|
    t^{j-k}e^{-\frac{t^2}{4}}L^{j-k}_k(\frac{t^2}{2}).\end{aligned}$$ where $\tilde{f}_{k,j}$ denotes the Fourier transform of the Pattern function $f_{k,j}$.

Let us state the lemmata which are used to prove upper bounds in Propositions \[prop:3\], \[prop:4\] and \[prop:5\].

\[lm:4\] There exist constants $C_2$, $C_\infty$ such that

$\sum_{j+k=0}^{N} {{\left\lVert f_{k,j} \right\rVert}_{2}}^2 \leq C_2 N^{\frac{17}{6}}$ and $\quad\sum_{j + k = 0}^{N} {{\left\lVert f_{k,j} \right\rVert}_{\infty}}^2
        \leq C_\infty N^{\frac{10}{3}}$.

This is a slight improvement over .

By symmetry we can restrict the sum to $j \geq k$. For fixed $k$ and $j$ we have $$\begin{aligned}
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{2}}^2 &=& \int_{{{\left\lvertt\right\rvert}}<2s}
        {{\left\lvert\tilde{f}_{k,j}(t)\right\rvert}}^2 dt
        + \int_{{{\left\lvertt\right\rvert}}>2s} {{\left\lvert\tilde{f}_{k,j}(t)\right\rvert}}^2 dt
    \end{aligned}$$ (with $s = \sqrt{k+j+1}$). Because of Lemma \[lemm:estimeLaguerre\], it is clear that the second integral is negligible in front of the first one, which we simply bound by $4 s {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2$.

In view of (\[pattern\]), the main result in [@Krasikov:2007uq] can 