Soit \(E = \R_{2n-1}[X]\), et \(x_1,\dots,x_n \in \R\) distincts. On note : \[{\phi_i} : E\to \R, P \mapsto {P(x_i)} ; \qquad {\psi_i} : E \to \R, P \mapsto {P'(x_i)}\]
1
Montrer que \((\phi_1,\dots,\phi_n,\psi_1,\dots,\psi_n)\) est une base de \(E^*\).
2
Chercher la base duale. On notera \(P_i = \prod_{j\ne i}\frac{X-x_j}{x_i-x_j}\) et \(d_i = P_i'(x_i)\).