Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by $$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$ We henceforth admit that $g$ is of class $C^{1}$ on $[-1,1]$. Using integration by parts, show the existence of a real number $C$ such that $$\forall n \in \mathbb{N}, \quad \left|\int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x\right| \leqslant \frac{C}{2n+1}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
We henceforth admit that $g$ is of class $C^{1}$ on $[-1,1]$.
Using integration by parts, show the existence of a real number $C$ such that
$$\forall n \in \mathbb{N}, \quad \left|\int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x\right| \leqslant \frac{C}{2n+1}$$