Let $\mathbb{T}=\mathbb{R}/ \mathbb{Z}$ and let $I$ be an interval contained in $\mathbb{T}$. Let $\chi_I$ denote the characteristic function of $I$. Then why is the following true:
$\vert \hat{\chi}_I(k)\vert \leq \frac{1}{\pi\vert k \vert }$,
for every $k\in \mathbb{Z} \setminus \{0\}$, where $ \hat{\chi}_I(k)=\int_{\mathbb{T}}\chi_I(x) e^{-i2\pi k x} dx $? I see that there is a trivial change of variable inside the integral, but I always get a larger upper bound.