Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by $$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$ Establish the existence of a real number $D$, independent of $x$ and $t$, such that $$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by
$$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$
Establish the existence of a real number $D$, independent of $x$ and $t$, such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$