Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$. Show that $m = 2n+1$ if and only if the $x_i$ are the roots of $p_{n+1}$.
Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as
$$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$
where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $m = 2n+1$ if and only if the $x_i$ are the roots of $p_{n+1}$.