For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$. a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$. Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.) b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that
$$\sup_{x \in [-1,1]} |P(x)| \leq n$$
(Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have:
$$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$