of noise, it is necessary to adapt the pattern functions as follows. From now on, we shall use the notation . When $\frac12 < \eta \leq
1$, we denote by $f^\eta_{k,j}$ the function which has the following Fourier transform: $$\label{eq:patterneta}
    \tilde{f}^\eta_{k,j}(t) {\mathrel{\mathop:} = }\tilde{f}_{k,j}(t) e^{\gamma t^2}.$$ When $0< \eta \leq \frac12$, we introduce a cut-off parameter $\delta >
0$ and define $f^{\eta,\delta}_{k,j}$ via its Fourier transform: $$\label{eq:patternetadelta}
    \tilde{f}^{\eta,\delta}_{k,j}(t) {\mathrel{\mathop:} = }\tilde{f}_{k,j}(t)e^{\gamma t^2} \mathbb{I}{{\left( {{\left\lvertt\right\rvert}}\leq
    \frac1\delta \right)}}.$$

Then we compute bounds on these pattern functions.

\[lm:5\] For $1>\eta>1/2$, there exist constants $C_2^\eta$ and $C_\infty^\eta$ such that

$\sum_{j + k = 0}^{N} {{\left\lVert f^\eta_{k,j} \right\rVert}_{2}}^2 \leq
        C_2^\eta N^{\frac56} e^{8\gamma N}$ and $\quad \sum_{j + k=0}^{N} {{\left\lVert f^\eta_{k,j} \right\rVert}_{\infty}}^2
        \leq C_\infty^\eta N^{\frac{1}{3}} e^{8\gamma N}$.

The proof is similar to the previous one and we skip some details. Once again we assume $j \geq k$ and write $$\begin{aligned}
        {{\left\lVert \tilde{f}_{k,j}^\eta \right\rVert}_{2}}^2 &=& \int_{{{\left\lvertt\right\rvert}}<2s}
        {{\left\lvert\tilde{f}_{k,j}(t)\right\rvert}}^2 e^{2\gamma t^2}    dt
        + \int_{{{\left\lvertt\right\rvert}}>2s} {{\left\lvert\tilde{f}_{k,j}(t)\right\rvert}}^2
        e^{2\gamma t^2} dt
    \end{aligned}$$ (where $s = \sqrt{k+j+1}$). Because of Lemma \[lemm:estimeLaguerre\], the second integral is of the same order as the first one, which we bound by $${{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2 \int_{{{\left\lvertt\right\rvert}} < 2s}
        e^{2\gamma t^2}  dt \leq C
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2
        s^{-1} e^{8\gamma 