We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$ 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 $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. Show that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by:
$$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
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 $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Show that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.