Let $I = [ - 1,1 ]$, $\alpha > 0$, and $$f _ { \alpha } : \begin{array}{ccc} {[-1,1]} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{\alpha^2 + x^2} \end{array}.$$ For $n \in \mathbb{N}^*$, let $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, and let $R_n \in \mathbb{R}_{2n-1}[X]$ be the interpolating polynomial of $f_\alpha$ at the $2n$ real numbers $\{\pm a_{k,n} \mid k \in \llbracket 0, n-1 \rrbracket\}$. Set $Q_n(X) = 1 - (X^2 + \alpha^2) R_n(X)$. Show that $R_n$ is an even polynomial and determine $Q_n(\alpha \mathrm{i})$.
Let $I = [ - 1,1 ]$, $\alpha > 0$, and
$$f _ { \alpha } : \begin{array}{ccc} {[-1,1]} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{\alpha^2 + x^2} \end{array}.$$
For $n \in \mathbb{N}^*$, let $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, and let $R_n \in \mathbb{R}_{2n-1}[X]$ be the interpolating polynomial of $f_\alpha$ at the $2n$ real numbers $\{\pm a_{k,n} \mid k \in \llbracket 0, n-1 \rrbracket\}$. Set $Q_n(X) = 1 - (X^2 + \alpha^2) R_n(X)$. Show that $R_n$ is an even polynomial and determine $Q_n(\alpha \mathrm{i})$.