grandes-ecoles 2023 Q33

grandes-ecoles · France · centrale-maths1__official Sequences and Series Properties and Manipulation of Power Series or Formal Series
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Deduce that, for every shift-invariant endomorphism $T$, we have $$T = \sum_{k=0}^{+\infty} (T q_k)(0) Q^k$$
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.

Deduce that, for every shift-invariant endomorphism $T$, we have
$$T = \sum_{k=0}^{+\infty} (T q_k)(0) Q^k$$