Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $Q_n(X) = 1 - (X^2 + \alpha^2)R_n(X)$. Show that there exists $\lambda_n \in \mathbb{R}$ such that $$\forall x \in [ - 1,1 ] , \quad Q _ { n } ( x ) = \lambda _ { n } \prod _ { k = 0 } ^ { n - 1 } \left( x ^ { 2 } - a _ { k , n } ^ { 2 } \right).$$
Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $Q_n(X) = 1 - (X^2 + \alpha^2)R_n(X)$. Show that there exists $\lambda_n \in \mathbb{R}$ such that
$$\forall x \in [ - 1,1 ] , \quad Q _ { n } ( x ) = \lambda _ { n } \prod _ { k = 0 } ^ { n - 1 } \left( x ^ { 2 } - a _ { k , n } ^ { 2 } \right).$$