grandes-ecoles 2022 Q7

grandes-ecoles · France · centrale-maths2__official Factor & Remainder Theorem Lagrange Interpolation and Basis Representation
Show that there exist $K$ polynomials $L_1, \ldots, L_K$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^K([0,1])$, the polynomial $P = \sum_{j=1}^K f\left(x_j\right) L_j$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_\ell\right) = f\left(x_\ell\right).$$ 
Show that there exist $K$ polynomials $L_1, \ldots, L_K$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^K([0,1])$, the polynomial $P = \sum_{j=1}^K f\left(x_j\right) L_j$ satisfies
$$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_\ell\right) = f\left(x_\ell\right).$$
Paper Questions
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