For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. Let $f \in C^\infty([-1,1])$. Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that $$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$ for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.
For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function:
$$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Let $f \in C^\infty([-1,1])$.
Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that
$$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$
for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.