Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$. a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$ b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$ c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$ d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
Let $T$ be a trigonometric polynomial of the form
$$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$
where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that:
$$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have:
$$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that:
$$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that:
$$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$