Show that, for any real $\xi$, there exists a real sequence $\left(c_{p}(\xi)\right)_{p \in \mathbb{N}}$ such that
$$\forall x \in \mathbb{R}, \quad \exp\left(-x^{2}\right) \cos(2\pi \xi x) = \sum_{p=0}^{+\infty} c_{p}(\xi) \exp\left(-x^{2}\right) x^{2p}$$