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 $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_\ell\right) = f\left(x_\ell\right).$$