The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ More generally, for $j \in \llbracket 1, n \rrbracket$, show the following equalities: $$\operatorname{ker}\left(\delta^j\right) = \mathbb{R}_{j-1}[X] \quad \text{and} \quad \operatorname{Im}\left(\delta^j\right) = \mathbb{R}_{n-j}[X]$$
The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$:
$$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$
More generally, for $j \in \llbracket 1, n \rrbracket$, show the following equalities:
$$\operatorname{ker}\left(\delta^j\right) = \mathbb{R}_{j-1}[X] \quad \text{and} \quad \operatorname{Im}\left(\delta^j\right) = \mathbb{R}_{n-j}[X]$$