s^2}.$$ In the sum we are considering the terms $k \geq 35$ and $j-k \geq
    24$ are dominant and, once again thanks to (\[eq:6\]), remembering that $s = \sqrt{j+k+1}$, $${{\left\lVert \tilde{f}_{k,j}^\eta \right\rVert}_{2}}^2 \leq C
        k^{-\frac16} e^{8\gamma (j+k)}$$ hence the first inequality.

The second inequality is, in the same fashion, based on $$\begin{aligned}
        {{\left\lVert {f}_{k,j}^\eta \right\rVert}_{\infty}}^2 \leq
        {{\left\lVert \tilde{f}_{k,j}^\eta \right\rVert}_{1}}^2 &\leq& C {{\left( j^{\frac14}
          k^{-\frac1{12}} \int_{{{\left\lvertt\right\rvert}} < 2 s}
          e^{\gamma t^2} dt \right)}}^2 \\
        &\leq& C j^{-\frac12} k^{-\frac16} e^{8\gamma(j+k)}
    \end{aligned}$$ when $k \geq 35$ and $j-k \geq 24$, and the bound on the sum readily follows.

Estimation procedure
--------------------

For $N {\mathrel{\mathop:} = }N(n) \rightarrow \infty$ and $\delta {\mathrel{\mathop:} = }\delta(n)
\rightarrow 0$, let us define our estimator of $\rho_{j,k}$ for $0
\leq j+k \leq N-1$ by $$\label{estrhojk}
    \hat{\rho}^\eta_{j,k} {\mathrel{\mathop:} = }\frac{1}{n}
    \sum_{\ell=1}^{n}G_{j,k}{{\left( \frac{Y_\ell}{\sqrt\eta},\Phi_\ell \right)}},$$ where $$G_{j,k}(x,\phi) {\mathrel{\mathop:} = }\left\{ \begin{array}{ll}
                                 f^\eta_{j,k}(x) e^{-i(j-k)\phi} &{\rm \ if \ } \frac12 < \eta \leq 1 \\
                                 f^{\eta,\delta}_{j,k}(x) e^{-i(j-k)\phi} &{\rm \ if \ } 0 < \eta \leq\frac12.
                                       \end{array} 
                               \right.$$ using the pattern functions defined in (\[eq:patterneta\]) and (\[eq:patternetadelta\]). We assume that the density matrix $\rho$ belongs to the class $\mathcal{R}(B,r)$ defined in (\[eq.classcoeff\]). In order to evaluate the performance of our estimators we 