We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. For a function $h \in C([-1,1])$, $\widetilde{h}$ denotes the $2\pi$-periodic function $\theta \mapsto h(\cos(\theta))$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
The purpose of this question is to show that the function $f$ is of class $C^\infty$.
a) Let $k \in \mathbb{N}^*$. Show that, for $P \in E_{k-1}$, $a_k(\widetilde{f}) = a_k(\widetilde{f-P})$.
b) Deduce that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ of Fourier coefficients of the function $\widetilde{f}$ has rapid decay.
c) Conclude.