Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$. We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$. Show that there exists an element $F_{r}$ of $\mathcal{R}_{n}$ whose poles are all in $\mathbb{D}_{1/r}$ and such that the following two properties are satisfied: $$\begin{gathered}
\forall z \in \mathbf{C} \backslash \mathbb{D}_{1/r}, \quad \left|F_{r}(z)\right| \leq \frac{b^{\prime}(M)}{r|z|-1} \\
\forall k \in \mathbf{N}, \quad X^{T}M^{k}Y = \frac{r^{k+1}}{2\pi} \int_{-\pi}^{\pi} F_{r}\left(e^{it}\right) e^{i(k+1)t} \mathrm{~d}t
\end{gathered}$$
Let $M \in \mathcal{B}_{n}$. We define
$$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$
where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
Show that there exists an element $F_{r}$ of $\mathcal{R}_{n}$ whose poles are all in $\mathbb{D}_{1/r}$ and such that the following two properties are satisfied:
$$\begin{gathered}
\forall z \in \mathbf{C} \backslash \mathbb{D}_{1/r}, \quad \left|F_{r}(z)\right| \leq \frac{b^{\prime}(M)}{r|z|-1} \\
\forall k \in \mathbf{N}, \quad X^{T}M^{k}Y = \frac{r^{k+1}}{2\pi} \int_{-\pi}^{\pi} F_{r}\left(e^{it}\right) e^{i(k+1)t} \mathrm{~d}t
\end{gathered}$$